I confess I do this -- not because I can't get the bike to stay upright (although having said that, the size/proportions of FCC entrances are a real PITA) -- but because I have to move the bike from one side of the carriage to the other at each stop so that people can get in or out. But people don't look where they're going -- use the door at the other end of the carriage?? -- and if I don't move fast enough, bad things can happen. Ever had someone lean over your bike, push the 'open' button and then just shove your bike out of the carriage?? I've seen it happen.
Err, no, because I change it before the train arrives..... In fact, immediately after the previous stop I change it so that by the time the train gets to the offending station, nobody is in the way of anything. And I don't sit halfway down the train from it (see below), I stay close to act as quickly as.
(Oh and it won't fall over either.

)
If you do it the right way round then it won't anyway! I have been doing it for years and can confirm this. There is a certain way of doing it so that the bike is pushing against the forward motion of the train if that makes sense. Also the bike gets wedged in to the doorway/vestibule so the rubber of the tires also help with adhesion. When I do it, it falls down on average every 7 years! Ha ha!
Anyway, if you know the route, change it over before you get to the station, then nobody can do anything, it isn't difficult. I prefer to sit near to the bike so I can act quickly, so even on a new route, you can at least judge certain stops, although you have still got to be ultra alert.
I know the not knowing the route is an excuse for holding your bike, I get it, but I seem to have made an entire art form of doing the complete opposite!
In fact, the only times I stand with the bike is when I can't get a seat, i.e. a busy rush hour train, or at least a seat near enough - you don't sit halfway down the coach from it unless you know it is ok, but if I can sit down near to the bike, then I will.
On the train from Dumbarton to Glasgow Queen Street there are 3 stations where you have to change sides (There are 6 if you go to Glasgow Central, but 3 of them are in a row, including Glasgow Central, so you just need to change it once.) It isn't exactly hard work considering there are many times that amount of stops where the platform is on the other side.
It just means you don't have to worry about it and can relax, and yet you are alert enough to always know where you are.
How can I possibly relax and properly read the paper/study/eat/veg out when I'm holding my bike?
Sorry, with all due respect, it just seems like the simplest thing to me and I feel like I'm the only person in the universe who actually does it. Of course, I don't travel in London, but I am talking of my fellow Glaswegian travellers who, it seems, have never ever worked out what I do!
I just leave people alone now - if they want to knacker their arm needlessly holding their bike the entire way, then let them, although don't expect much sympathy from me.
In fact, the only thing worse than needlessly holding your bike is your bike taking up half the aisle!
I remember a woman doing that once and the stupid bint tisked, gave dirty looks and moaned to everyone who had the cheek to actually want to get past, including me. To be fair, it was a lovely Pashley Princess, but, still, I was far from impressed by the bikes owner.
I give up. Cyclists on trains (except for me of course) put 'em all into room 101!! Muhahahaha!!