This can go two ways:
- Nobody notices, or they notice but nobody minds. You get the money.
- Somebody notices, and it turns out your boss: had no authority to hire you for the work without approval; worst case would never have got permission to hire you for the original work; therefore is knowingly defrauding the University. You're both involved in criminal fraud to the tune of 300 hours of pay. Perhaps your boss even denies ever having had this conversation with you, in the hope you'll go down for it alone.
So the worst-case potential downside is prison. Probably the money's not worth the risk, but your call.
Granted, the worst case is unlikely, and other people are working around the bureaucracy in similar ways. Still, you want to avoid doing anything shady, and this is shady because the people paying the money aren't being told what it was for. That justifies pressing your boss to admit to HR and his boss that he was slack about the paperwork, and request that you be paid for all or some of the work you've already done. It's much better to submit a truthful invoice for the work you've done, than a bunch of false timesheets for which you can be held liable, maybe even years from now. If the University refuses to pay the invoice on grounds all you had was a verbal agreement with your immediate boss, you can either let it go or you can explore other options to demand payment.
If all that's going on here is your boss doesn't want to have to confess to being badly-organized, then it might not even be that difficult to convince him to face the music rather than do something that probably isn't legal even if it does "come out the same in the end". Really depends on his personality and the consequences for him of anyone finding out what he already did wrong (strung you along for 300 hours of unauthorised work). If he's already committed a sacking offence then you can only choose two of getting paid, staying legal, and continuing to work for him. You couldn't have all three. If he's merely at risk of looking like a buffoon and getting frowned at then maybe he's willing to do that in preference to involving you in something you're not happy with, and you get all three.