Bail is actually more complicated than usually shown on TV. The Wikipedia page shows 9 different kinds (and a 10th "Any combination of the above"). Not all of them are relevant to all jurisdictions, but "One billion dollars bail" could really be several forms (my comments in italics after each one):
2) Unsecured bail. It is still a release without a deposit but it differs from [released on recognizance]in that the defendant must pay a fee upon breaching the terms of the bail. Since no money changes hands, this wouldn't help.
3) Percentage bail. The defendant deposits only a percentage of the bail's amount (usually 10%) with the court clerk. This would still be $100,000,000 on the billion, which would be a noticeable sum.
6) Property Bond The accused or a person acting on his behalf pledges real property having a value at least equal to the amount of the bail. If the principal fails to appear for trial the state can levy or institute foreclosure proceedings against the property to recover the bail. Used in rare cases and in certain jurisdictions. Often, the equity of the property must be twice the amount of the bail set. For a person who owns two billion dollars of property, this might be an option, but since it isn't liquid, the city wouldn't be able to use it until it was foreclosed and sold.
8) Cash Typically "cash-only," where the only form of bail that the Court will accept is cash. Court-ordered cash bonds require the total amount of bail to be posted in cash. The court holds this money until the case is concluded. This is almost certainly the form you're thinking of.
So, assuming that the cash or percentage options are what you're thinking of, is it feasible for the city to "float" this money? Not really.
Generally, the money is held by the court (not the city), so getting access to it for other purposes would be a problem. Additionally, the court has to be ready to return the money at any time - for example, if the charges are dropped or the accused dies, the bail will be returned. This has been researched a bit by the people in this thread, where the argument boils down to "Bail is to make sure you appear. If you no longer need to appear or you can't appear, then there's no need to hold it and it's returned."
This makes using it for any purpose very dangerous, because you might be suddenly expected to have that lump sum in hand - and if you're using it to "float" payments, you explicitly don't have that much around. Given the typical "at least 2 weeks" lead time government refunds can take, the city wouldn't necessarily need the cash in the bank right at that very moment, but if it suddenly had to re-raise that much money to repay you in a short period of time, it would be just as bad.
In short, it's effectively impossible, highly unlikely to even try and get around the impossibility, and it would be a really stupid idea for a city to try.