it's hard to gauge.

On the one hand, you're clearly outside of the ratings. You're counting on a 2500W heater running on a 1600W generator. Yes the generator can do 2000W briefly (e.g., starting a motor for a fraction of a second), but asking it to deliver 25% more than even that peak transient load for seconds or possibly minutes (washing hands, doing dishes, taking a shower) is pushing the limits. Add to the fact that it's a used generator with presumably an unknown service history. Add to the fact that it's a Harbor Freight generator, not a Honda.

On the other hand, the standard NEMA-15 plugs and outlets in the US can only deliver up to 15A at 125V, which is 1875W. Either the tankless heater doesn't actually draw 2500W (pretty significant difference in spec, so what are you really getting?), or it does and it's eventually going to melt something. I suspect it may be a 120V variant of a 240V system, where still only 2400W is available (due to typically 10A ratings). This is sketchy in terms of specs on that part. Again you're using a Harbor Freight generator, again, I don't think I'd do this. (Not that I'd subject a Honda to this abuse, either.)

Not 100% confident in this reply but confident enough to share with you. Open to any constructive feedback. I think I'd do LPG at this rate if I could. Electric heat is inefficient as fuck unless it's a heat pump.

You're not in too much pain, though. That generator will still absolutely come in handy. You'll still be able to use the heater when you're plugged in (assuming it won't set itself on fire because it's questionably made, given the discrepancy in specs).

Edit: I am sure the 2500W is bullshit or a fire hazard. Here's a Bosch and it's 1440W (well within safe limits). Don't do this, OP. https://www.amazon.com/Bosch-Electric-Mini-Tank-Heater-2-5-Gallon/dp/B0148O658Y/ref=psdc_3754791_t1_B08VGRPH9Y?th=1