Nothing bad will happen in bonding both poles of your panel to the same 120 volt phase provided that it is limited to 15 amps.
I have two generators (and there's a long story behind that) where one has a maximum of 2000 watts and the other 6500 watts. The smaller generator outputs 120 volts and to attach that to the house for power I bond the two phases. The larger generator outputs 240 volts, two 120 poles out of phase like the utility power. Perhaps I'm pushing the limits a bit on the smaller generator as it could provide a bit more than the 15 amps that would otherwise trip an overload on a MWBC but if there's so much as 2 amps somewhere else on that 200 amp panel there's going to be a trip on the generator before there is damage to the wires.
Also, don't forget that it is not like a 15 amp breaker is fine with 14.9 amps and then trips instantly if that spikes to 15.1 amps. There's a variation on the trip levels among the breakers based on manufacturing, wear, and so on. This is in part why there is that 80%/125% rule for heating loads. The trip current on a 15 amp breaker is not likely to be exactly 15 amps, it is going to be some small margin above that to prevent nuisance trips but not so much higher that safety isn't maintained by letting any wires or contacts get too hot.
I expect that any generator that gets approval from UL, DOE, FCC, EPA, ATF, FAA, WTF, BBQ, and whichever other TLA that comes along will have a 120-0-120 output on a 4 prong outlet if it was sold in the USA and has an output greater than is safe from a NEMA 5-15R. Edit to add: I realized later that by being so emphatic I've crossed the line into hyperbole so I'll clarify this in my summary at the end.
Any 240 volt load on the panel will see zero volts. Given that there could be some small difference in resistance of the electrical path between where the panel has both poles bonded and the various loads branch off there could be some small voltage showing. This would hardly damage the large motors and resistance heaters seen using 240 volt supplies to operate, they'd likely see voltages like that from all kinds of weird induction and such.
A 240 volt load like an air conditioning unit might have 3 wires to it but the ground wire is not (or should not be if the thing is still intact) part of the circuit, any voltage difference between that and either pole (within reason) does not matter. On one pole is one end of a very sturdy electromagnet, heating element, or something, and on the other pole is the other end of this very sturdy piece of equipment. To get the high power flow through these devices there's a low resistance, and if there's someone trying to turn these things on while the two 120 volt poles are in phase the voltage will only get lower. The lower the voltage the less current, the less current the less power. That voltage I refer to is, again, the voltage between the two poles and not the 120 volt input and so will be very very low, as in fractions of a volt. Some current might flow but it would be very small, not enough to make any motor move and not enough to produce any meaningful heating.
If the total current input is limited to less than 15 amps by a circuit breaker at the generator then even MWBC circuits will operate safely. Given some margins for error, "phantom loads" around the house sucking up a few amps, and so on I'd think even a 20 amp or 25 amp breaker is safe. But then what I consider safe for my house might not fly for you and your insurance company so it's on you if taking my advice breaks something. I do want to spell out why bonding the poles is considered a safe practice and done routinely. Generator power is a temporary situation, done in an emergent situation, so there's going to be some safety rules that get set aside. Do know when you are breaking the rules, why one rule overrides the others, and what is at risk if there's another emergent situation that comes up to cause a reevaluation of the risks.
With a generator producing 120 volts at a maximum of 15 amps there's no issues with 240 volt loads or MWBC, the circuit breaker (or GFCI if it has one) on the generator should trip before anything in the load panel trips.
Edit to add: For those that believe I was hyperbolic before I'll emphasize again my important caveat on what would make this safe, there's a 15 amp breaker at the generator. Also, there's a at least a half dozen reasons to argue this is safe if the breaker at the generator is 20, 25, or maybe even 30 amps. The important detail is knowing why bonding the 120 volt poles could cause a safety issue, by knowing the cause there's ways people can mitigate against any hazard. If you don't know when and why bonding the two 120 volt poles on the breaker panel is safe then it may be wise to not do so. If you know why it is not safe then that will inform you on how to make it safe.