Air carrier First Officer reported possible loss of separation during climb with corresponding TA alert. Flight continued climb.

Narrative

On Taxi-out for departure from runway 09 at EYW; ATC advised us to make a left turn heading 360 on departure. We were then cleared for takeoff from runway 09. As PF; after passing 400' AGL; I requested heading select (360) and proceeded to start a left turn to heading 360 flying with the autopilot off. While making the turn and scanning outside the aircraft; I immediately saw a group of targets that I initially suspected might be very large birds. At some point during this turn EYW tower informed us of traffic on a left base into Navy Key West and directed us to contact departure. My initial instinct upon seeing what I now recognized as opposite direction air traffic (a flight of 4 military aircraft) was to continue my climbing left turn in an effort to see and avoid what I felt was a potential traffic conflict. At about the same time I recognized a TCAS traffic alert and noted a target -300' (or 300 feet below us). I believe this occurred while we were at approximately 700' AGL due to the fact that we received an audible traffic alert (which is inhibited below 600' AGL). With the traffic visually acquired and verified on TCAS; I rolled out on our assigned heading and continued our climb satisfied that we were no longer in a potential conflict. During this time the Captain acting as PM; checked in with Navy Key West departure and queried them about the traffic and procedures for deconfliction since we were not given any information prior to beginning our takeoff roll. ATC simply stated that the two airports are extremely close to one another and that the military aircraft were maintaining visual separation with us. Our flight continued on without further incident.I truly feel that if we were at a higher altitude; we would have received a TCAS Resolution Advisory (RA) due to this traffic. According to our manual; RA's are inhibited below 1100' AGL. Since this all occurred at approximately 700'; I believe we were too low for the aircraft to command a RA and instead simply announced an audible 'traffic; traffic' alert. Cause: I believe the EYW tower's failure to alert us to the traffic inbound to Navy EYW contributed to a possible loss of safe separation between our aircraft and the military aircraft. At a minimum it should be required to alert aircraft departing from runway 09 at EYW to traffic inbound to Navy Key West so that crews can factor it into their decision making process. It would be better for ATC to just delay IFR departures until VFR traffic inbound to Navy EYW is completely clear of the departure corridor. Simply relying on military aircraft maintaining visual separation without our knowledge is not an appropriate solution to this kind of situation. This event occurred exactly a day prior to the anniversary of a tragedy in the industry that in my opinion it had striking similarities. A one-minute departure delay to allow the military aircraft to clear the departure corridor and land would have been appropriate to mitigate this potential threat.It would also be helpful to include a note in our company pages about the potential for this kind of traffic encounter; so that crews can include it in their threat forward briefing prior to departure. Had we known about the inbound traffic; we could have made the decision to delay our takeoff in an effort to mitigate this threat.