I would suggest you use ffmpeg and call it from PHP using shell_exec() for example.
If you use x264 for encoding you can tune it to use moderate/lower encoding settings so that your server can deal with the load (up to a certain point of course in a size-controlled environment).
ffmpeg will also allow to trim (and much more) videos as you see fit. Video hosting services may not have that option or this could induce an additional cost.
Beyond that recommending online video services to fit your project does not fit Stackoverflow guidelines for asking questions.