Yes, I’ve moved transponders between cars without registering license plates and it worked fine. I’ve also had a transponder go bad, so having my license number on file helped - flat rate charge instead of a citation/fine.
That was after about 10 years of usage - the battery died. The odds of your transponder failing are very low. If you can, try it once and see that it’s working, then don’t worry about rental cars. (Or add it just to be safe - not that hard).
Side note - in PA it’s a flat rate toll if you’re registered but the transponder isn’t working. The one that failed for us was on my D’s car, who uses it when she’s home from school - no toll roads - and the one we pack up for trips to the in-laws, which is 3 hrs on the PA turnpike. So we saved $15 every time we used it. I took my time replacing it once I found the issue. Which was nearly 2 years later - they don’t tell you when you get a flat rate charge. I imagine that if you commuted one exit every day it could get quite expensive.