Well they can always send a routine congressional inquiry, i dont know why your congressperson is so stingy with a simple service. Shop around till you find a congressperson or senator willing to do this for you. But its true that uscis will just respond in 30 to 45 days with a boilerplate answer that they are working on your case. unless they sent you rfe or noid and you didnt get it, in that case they will tell congressperson that they sent RFE / NOID. For example my lawyer misplaced RFE for medicals and i never got it, only found out when USCIS told senator that they sent RFE and were waiting for my response. They never updated online portal for the RFE either. So that is one thing - information, that is advantage of a congressional inquiry.
Mandamus - completely depends on the judge you get, but usually any petition beyond two years the judge will frown at uscis and may be more likely to rule in your favor if the case goes to court. I think that is probably why VSC limits the processing time to as close to two years as possible, they know they will have a slew of court cases if they make their "posted" processing times more unreasonable than that. (criticism welcome from forum, this is just my opinion.) More importantly USCIS hates going to court, as that is a lot more work for them. They will likely approve your case faster if you file a mandamus at the two year mark. See @RKE 's post a few pages ago. They filed mandamus after about three years and got to green card in a few months. Now whether you want to file mandamus now or later is completely up to you. But it seems rational to wait the full 26 - 27 months for USCIS to get off their voluminous posteriors and adjudicate your I360.
https://www.jeelani-law.com/services/writ-of-mandamus/
https://www.hooyou.com/i-485/mandamus.html#:~:text=Sending a “Notice of Intention,evidence surrounding the potential lawsuit.