A topical antibacterial might help, though I’m sure you’ve tried it. I mention it because others reading might not have. For me, rapa-caused bacterial bumps can be treated with metroninidazole.
I do understand that mosquito bites are a different matter; I get swollen and inflamed mosquito bites even without rapa. Lifelong challenge. Finally found a fix. I’ll tell you what I do, in case it might help: heat. I take a folded rag edge, or sponge tip, or a used tea-bag, and apply that hot wet spot directly to the bump. If you do it right – there’s a small sweet spot between burning the skin and it not being hot enough --you should feel it get really itchy for a moment, then fade.
May well be a different mechanism of action than would address bacterial, but I wonder if it might help without the systemic downsides of doxycycline. Perhaps worth a try.