Disclaimer: I'm not a brass player... but just a thought - this reminds me of several threads I have read about people learning the oboe. Because the oboe requires a tiny volume of air but with masses of support and air volume behind it; the player has to breathe in a lot but only uses a tiny proportion of that breath to actually make the sound. At the end of phrases they HAVE to breathe out to get rid of stale air before they take another breath, or they risk fainting or dizziness from lack of oxygen. I wouldn't imagine it's as great an issue on the trumpet as on oboe, as you're not blowing into such a tiny space as with playing the oboe, but it might cause similar issues...?
It may be that from playing your other instruments you have a really good lung capacity and playing the trumpet is not ridding you of all the air in your lungs by the end of a phrase. (It's been posted on here that flute uses more air than the tuba due to wastage: though I think sax/clarinet/bassoon probably waste less air, maybe they need more than the trumpet does so your lung capacity is good? I'm guessing here!)
Forcing your body to breathe in when it's telling you it wants to breathe out doesn't sound very healthy

maybe that's just me. If you can find out how much air trumpet requires relative to the instruments you already play you might find that's the problem? Or maybe part of it? It seems worth investigating.
& at the risk of sounding totally dumb - have you tried playing longer phrases?? Though I guess you may end up not having enough air in your lungs to support the sound, I don't know...
Sorry, this is coming from a non-brassist's perspective, but can't help thinking it'd be worth looking into. If you know any woodwind/brass doublers you could see if they have experienced similar problems, or ask around oboists and see if their experience rings any bells for you. I may be talking complete tosh, so don't act on this unless by research you find that it is what's going on, but it does sound a lot like the experience oboists have from the descriptions I have read. You may discover it's something else entirely, of course! A good teacher and maybe one who has experience of teaching doublers would be a massive help to you.
Good for you for sticking with it - I really hope you find a solution.