Tower Local Controller reported a student pilot reported a NMAC with a banner tow aircraft.

Narrative

(1) ZZZ Approach hasn't called to APREQ multiple ODO (Opposite Direction Operations) arrivals while ZZZ is on east traffic. This leads to ZZZ1 not calling out ODO to the cab in a timely fashion until the aircraft is almost in our airspace; so at XA:31z I launch Aircraft X for a straight out off of RWY XXR. Once airborne I see on the scope they will conflict with a fast moving ODO arrival; so I ask if they're planning on going northbound or south to the practice area. Student pilot responds they'll go north; I approve the northbound turn into ZZZ1's airspace and after the readback I begin to coordinate with ZZZ1. While coordinating; I miss that the instructor on Aircraft X is in my ear saying they'll go south instead and they're turning south now to make room for our ODO. CIC (Controller in Charge) shouts out to correct my coordination with ZZZ1 that Aircraft X is going southbound. This is when I feel that the complexity is starting to build in the airspace and tension starting to rise.(2) Soon after (~XA:35z) I got a call from Aircraft Y to the southeast who I gave right traffic for RWY YYR; which was incorrect because of the runway in use was Runway XXR. When they asked to verify I said affirmative; because I was focused on making sure I had the spacing to depart Aircraft Z who had been waiting; while I was seeing where the still not called out ODO aircraft was; CIC called out that I said Runway YYR and I amended instructions to right traffic for Runway XXR. I was near the end of my OT shift at this point but feeling good about staying on for more seasoning time on ZZZ2 because the CIC was now focusing more attention to me and caught these two mistakes. I felt that I could take a breath; reset mentally; and finish the time on position with no more mistakes.(3) The wind begins changing and those of us in the cab start looking for an opportunity to change from RW XX's to YY's. A few things then happen in quick succession: ZZZ Approach misses an ODO coordination for someone they switched on the border of our airspace; but calls ZZZ1 at the same time as that aircraft checks in; about another aircraft 15 miles to the east; causing the first aircraft's transmission to be played over the loudspeaker. The aircraft on the border reaches out to ZZZ1 and appears to be about to turn away because they get no response from ZZZ1; who is still on the line with ZZZ Approach; and starts turning south towards my airspace. GC (Ground Control) advises me that we need to make some holes to taxi people across the departure end of Runway XXR because there will be too many planes between those currently on the ground and the inbounds for Runway XXL which will cause go-arounds from the north runway.(4) At around XA:37z Aircraft A which was on final for a full stop on my runway; goes around from the south runway. I didn't have time to ask the pilot why because while there was a strong tailwind at the time; Aircraft A and the other aircraft from the same flight school have regularly started going around as they cross the threshold to save money from the landing fees the city X have implemented. (It is cheaper to go around than to do a T/G or full stop/taxi back.) I jump on frequency much earlier than I usually would to Aircraft A- before they're 100ft above the runway- telling them to extend upwind and I will call their crosswind. There is now a small window to hit between the ODO aircraft inbound to circle north for Runway XXL and Aircraft Y who I have in the downwind already. Once Aircraft Y and Aircraft A are abeam each other; I call for Aircraft A to turn crosswind and advise traffic they're following is the sling in the downwind. At this point we need to do the crossings as well for the south runway; or the north runway will run out of room; so I extend both of Aircraft Y and Aircraft A's downwinds; telling them I'll call their base turns. Both aircraft read back correctly.(5) As I'm looking at the crossings trying to gauge the timing; I notice a target low level entering my airspace from ZZZ1's airspace. I don't recognize the callsign; but at 200 ft and 40 kts I assume it's a low-level banner tow making a southbound transition along the shore as they usually do; and because of the amount of airplanes and tasks to do; I did not have a chance to confirm with ZZZ1. I look back up from the scope to check the state of the crossings; then down again when I notice the banner tow is in a turn. The banner tow's tag was set to the 9 direction on the scope; and Aircraft Y was set to 1 on the scope. (Usually ZZZ2 moves their data blocks below the target and ZZZ1 has them above to provide a quick visual reference as to which targets belong to who.) This put both data blocks for Aircraft Y and the banner tow over each other and I started to use the slew ball to separate the tags since I couldn't read the altitude for either target now. CIC tells me to base Aircraft Y; so while manipulating the slew ball to move the tags I transmit that they should turn base; cleared to land Runway XXR; and the wind. The student pilot reads back the full clearance; not indicating any problem. At the end of the readback; someone on Aircraft Y keeps the transmit button held down and we hear 'Holy Crap' on frequency; followed by static as the button continues to be held down. I try to reach out to ZZZ1 to ask what the banner tow is planning to do in my airspace because I had not heard a point-out given. I have the tags separated at this point and see the banner tow is now northeast-bound turning southeast at the same altitude as my pattern traffic and inside half a mile from the sling. Aircraft Y's base turn and the banner tow's southeast turn take them on diverging courses; with there being resolution between the targets; so I reach out to Aircraft A who is (as best as I can remember while writing this) two miles in trail of Aircraft Y and now has ODO traffic in their face. I tell them about the 12 o clock traffic but get no response as Aircraft Y still has the button held down and is stuck-mic. Aircraft Y eventually releases the button and Aircraft A/the banner tow turn to pass each other.(6) ZZZ1 is still task saturated dealing with ZZZ Approach's ODO misses so it takes a moment before we find out the banner tow was cleared to go south then eastbound while remaining two miles south of the airport. The location of the incident was not two miles south of the airport even by the generous definition of making the two mile line be parallel to the runways instead of true east-west. Aircraft Y asks what was up with the banner tow and reports the near miss. Aircraft A reports that they're on the base even though I didn't call their turn; but seems to be no factor; so I clear them to land number two. When they've returned to parking CIC uses the cab phone to call the flight school and get information from Aircraft Y; and ZZZ1 brashers the banner tow then switches them to my frequency because they'll be in my airspace. When leaving the airspace the banner tow advises it'll be about an hour before we can expect the call.Overall the factors that led to the event:(1) East traffic; changing winds; and ZZZ Approach missing ODO appreq's increased the complexity and late or missed coordination.(2) I assumed the low-level slow-moving plane was a south shoreline transition and did not get the details from ZZZ1- instead I just assumed the path and that ZZZ1 missed a pointout due to dealing with ZZZ Approach ODO issues.(3) When CIC told me to base Aircraft Y I did so without thinking even as I was trying to separate the tags to make sure there wasn't a problem.Recommendation: The LOA says ZZZ Approach must appreq ODO operations; missing those calls lead to increased workload and complexity that snowballed. Because I did not hear a point out for the banner tow traffic; I believe that certain coordination should be done on the line to make sure it's clear what the traffic is doing; the volume in the cab and pilots calling over the loudspeaker made for a noisy environment.