How to gracefully renegotiate a verbally agreed offer after receiving second higher offer?

Experienced professional in a somewhat specialist field here.

I verbally agreed to an offer with Company A. The recruiter lowballed salary initially, and in a another conversation that caught me off guard, the recruiter pressured me into spitting out a counter to him, even though I tried to express that hadn't had a ton of time to prepare for that conversation. My verbal number at least was above my bottom line, but because it was a pressure situation, I didn't have the presence of mind to respond more boldly with a higher number than that. The recruiter had to go two or three levels above the role's manager to get approval for my proposed number already, with success, and they prepared a letter with that number that I have not signed yet.

Then a day later Company B sends me an offer about 10% higher shortly after, without me having had to negotiate anything yet or state a number to them.

My preference is to work for Company A, and they assure me I'm the top candidate for the job. Is there a strategy to get that higher at this point without putting the position at risk? Yes, I have seen a lot of advice online already like "be prepared for them to pull it or to look stupid if you end up taking it after they say no to further increases".

Thanks