Small aircraft pilot reported inadvertently entering Class C airspace just after takeoff while maneuvering to avoid collision with another aircraft that was entering the VFR traffic pattern at a nearby airport.

Narrative

I finished run-up and was preparing to take off from MLE on Runway 12 with a westbound departure. I made my radio call before takeoff. Aircraft X [was] taking off Runway 12; departing to the west. While taxiing onto the runway a call from Aircraft Y was made. Aircraft Y; 3.5 miles southeast of the field; will be entering the downwind for [Runway] 12. Takeoff roll was fine; climbing out runway heading it was very gusty / bumpy; I noticed I wasn't climbing all that fast and was watching the altimeter; hit an updraft and was quickly climbing at 1200 FPM. Climbing through 700 ft. AGL I initiated my left crosswind turn and radio call. During the turn Aircraft Y called and asked where I was. I responded with on the crosswind" while rolling wings level. Aircraft Y called back with; "I've got you in sight." That's when I noticed; out my right window 1 o'clock; Aircraft Y approximately 500 ft. away and 100 ft. above. At this point rolled into my left downwind turn and increased my rate of climb. Aircraft Y turned right and I believe descended; with the high rate of climb; and wanting to ensure I was clear of traffic. I entered the bottom of the OMA Class C outer ring. It is 2500 - 5000 and just off the departure end of Runway 12 at MLE."