Okay, that gives a couple ideas that "might" be involved. One is that some RV furnaces do have what they call a "lockout" after trying to light three times in a row. When you need to reset it, that kind of sounds like you are needing to reset the lockout.
On that point, do you have a furnace manual in a big bag of info that should have come with the RV? That might be the easy way to go for info on any reset your brand of furnace may have.
If manual not on hand, I think this may be one of the more common furnaces, so may match what you need?
The idea of the weak battery may be something to keep in mind if it continues to be a problem.
But there is one way to assure the battery supply is not going to be weak when starting the generator. Thinking of the starter as using a lot of power suddenly when we start any engine, if we think it "might" be a weak battery situation, one way to do away with that question, is to start the RV engine first.
Without knowing which year and model of Viasta, we can't look at drawings to say for sure what system you have but all Winnebago I've looked at had a system to tie the coach batteries and chassis/start batteries together as we drive. Some have a mode solenoid, while newer have a BIM to do this.
What this can do for sorting the weak battery question is that it connects coach and start battery as well as that big old alternator, so that the generator is sure to get plenty of power to start without dragging any battery down low enough to make the furnace funky!
So if the furnace continues to be a problem, maybe try starting the RV engine first, then then crank the generator. If that solves the need to reset the furnace, then you can begin to look closer at possible battery questions. Always keeping in mind that what may seem like a weak battery may actually be corroded/dirty/loose cables.
See if any of that helps and let us know how the guesses turn out?