"Introducing BI as a tiny fraction of GDP"
OK, yes, but what's going to keep it a tiny fraction of the GDP? The BI's natural "lobbying organzation" is the entire public... not even the entire voting public, but the entire public. No current structures have that characteristic. For almost the entire population, $10,000 is a lot of money. Even people making $150,000/year are going to still appreciate $10,000, and we're now talking well above median.
If there was a lobbying force large enough to actually countervail the entire public's interest in voting themselves more money, I'm pretty sure it would not be hard to drum up a mob against them.
As I understand it, Basic Income's distinguishing characteristic is that everybody gets it. You can't use our experiences with things that well less than half the population gets to judge the effect it will have on voting patterns.
You can't just think about the first year of Basic Income's implementation. You have to think about the second year. You have to think about the tenth year. You have to think about the 18th year, when people who have never lived in a regime other than having BI exist start voting themselves. Are they going to understand that it's a bad idea to listen to the politicians promising to make it larger? Because there will be such politicians. Probably all of them, honestly.