I don’t have a definitive answer for you, but here are a couple things you could investigate. Do you ever take your pulse along with your temp? For example, take your temp and pulse before you drink coffee. Then, 30 mins or so after having some coffee, check both again. If your pulse was 65 before coffee and goes up above 85 (say 90-100) after the coffee, the coffee might be causing problems. Not sure if you’re familiar with Ray Peat or the practitioners who follow him, but he thinks optimal pulses are 75-85 bpm. So the coffee could be activating your stress hormones if your pulse goes too high, and then the high temp might be faking you out because it’s also increased by the stress hormones.
Of course you probably have to take your pulse regularly to see what your usual bpm is– that way you can figure out whether certain foods or activities increase or decrease the pulse– a one-time measure after coffee probably won’t give you the full picture.
Another thing to consider: how much coffee are you drinking? It’s possible if you’re super sensitive, then an ounce or two could cause problems for you. But, it’s also possible that 4-8oz per day could give you a boost and actually help your metabolism. If I understand your post, you’re saying that you drink coffee and your morning temp the next day is in the low 97s? What if you cut coffee out for several days and then check your morning temp. If the morning temp after a coffee-free day (or after several coffee-free days) is higher (and you didn’t change anything else in your routine), then maybe coffee is a problem for you.
Also, if you’ve been sick or had a low metabolism for a while and you’re drinkg 16oz or more per day, it probably wouldn’t be a bad idea to back off the caffeine for a bit to see how your body and your temps react.
If coffee only seems to be doing positive things for you (doesn’t make you jittery, doesn’t upset your stomach, doens’t make you crash later, doesn’t send your pulse sky-high), then maybe it’s a good tool for you to use.
Also, if a full 8oz cup causes some bad symptoms for you, you could even consider sipping an ounce or so every hour (whenever works for you, say from 8am-1pm), instead of drinking 8oz all at once. Your body might handle the small load better and still get the metabolism boost from it.