Can any kind of immune response mess with BG levels?

I know being sick can elevate blood sugar, but can immune responses of any kind? I got my covid and rsv vaccines yesterday and i would say in the past couple of hours I've started feeling the covid shot (body aches, swollen lymph nodes, brain fog, etc). My readings from my extremely GD friendly breakfast (low carb, low sugar, high protein) gave me an elevated one hour reading of 144, well above where it usually is plus I'm on nighttime insulin for fasting which lowers my morning numbers as well. I'm assuming this is a response to the vaccine but also, want to give myself the extra reassurance lol. I was planning on taking it super easy today since i often don't feel amazing with the covid shot and i expected it to be a little worse this time given the pregnancy, but totally didn't think abt the blood sugar factor. anyway, if i get another elevated reading today i was going to give my MFM a heads up (idk why, I've been extra nervous abt escalating medical interventions even though that is absolutely not been the case with my doctors since this diagnosis), but has anyone else had this experience? In a normal circumstance, i would up my exercise a little, but tbh i kinda feel like garbage lol.