Or at least, if you're going to do this, you'd have to be very careful to not overcorrect for the 1st of tha month problem.
Not to ding your overall idea, I like it, think it's very cool. Just, to implement it with current infrastructure it would require a lot of care.
(Actually also I think rent or other large payments are less of a problem in your solution than you think - for anything on contract like that, you could simply reduce the rate at which continuous income is obtained. If your basic income is 100 units per whatever, and your rent is 43% of your basic income, you just reduce the stream down to 57 units)
Sure it does. Just save up for the 3 days before you go or whatever. There's nothing forcing you to spend every cent as it comes in, and a financially stable and responsible BI recipient would do well to save up, buy in bulk, and have a rainy-day fund anyway. Making the income into a stream would just save you from being kicked out on the street hungry if something unexpected happens and you hit 0 at some point.
Aside from what all else is going on in the society, everybody has a banked amount of money that is their steady-state; that amount is pegged to living costs (either civ wide, or locally, I don't remember, probably something in between). If you currently have less than that amount, your account gradually fills - if you current have more, it drains. I also don't remember if the rates are fixed, or are curved (further from baseline, higher the fill/drain rate is).
Time-based costs and payments - rent, salary - are then taken as a rate, while one-off payments (meals, events, items) are paid for normally.
The goal of the system is a more matured BI situation, /plus/ an incentive structure to keep producing. If you make a shit ton of money off something, it'll last for some amount of time, but you will have to eventually re-contribute to society in order to maintain the "rich" standard of living.
I think either of these solutions could be made to work in general. Certainly better than what we have right now.
There's actually nothing in this scheme that's tied to basic income either -- we could do this reform with existing entitlement payments today in a totally revenue-neutral way.