It's perfectly reasonable to say what you need in order to move jobs. Whether you reveal exactly why you need it is a tactic, according to whether you think the employer will agree to it otherwise.
One way of looking at this is that explaining why you literally cannot agree to move without the bonus reveals and proves your BATNA, which can be very good for you if your BATNA is bad for your opponent (that is, if they really want you to take the job and you won't take it without the money). Another way of looking at is is that your new employer will look more kindly on "covering your costs of switching jobs" than they will on "handing you cash for nothing".
Also, when you make it clear that the request is a demand, is a tactic. If you do it early then they might pass you up early without even considering you, because they know there are plenty of candidates who won't make that same demand. If you do it late then they're already invested in you and might be more willing to consider it seriously, then again they know you've played them, which is probably fine as long as they don't feel actively defrauded.
If you're correct that signing bonuses are common in your industry then they presumably won't feel defrauded if you spring such a demand on them late in the game. If you're wrong and they're uncommon for your role, they might feel the demand is unreasonable, you should have planned to take a personal loan for the money, and you've been wasting their time. Normally you want to reveal nothing until the salary negotiation at the end of the hiring process, but that changes if there's a risk they'll just reject what you need as ridiculous.
Presumably a signing bonus would in any case be on the same terms as the money you'll need to pay the old employer -- if you leave within X time then you pay it back. So indirectly they're just purchasing your indenture from your previous employer, which is fine if they are (in common with your old employer) comfortable with the indenture.
If signing bonuses and training paybacks are both common in your industry, then you can't be the first ever person in this situation, and there is probably a standard practice if only you can find someone to tell you about it. If so then following the standard will be seen by your new employer as totally professional, whereas not following it will be seen as either sloppy or malicious.