You have to go to the CHP to have the frame and engine numbers officially entered into the DMV system. DMV officially recognizes a form they fill out and sign. CHP should also run the numbers to see if they are on the stolen vehicle list.
You have a bill of sale (hopefully dated and with a traceable seller signature), to show you bought in good faith. This should relieve you of any charges if the bike is stolen. The bike will be confiscated if it is stolen along with any "improvements" you made.
Assuming it is not on the stolen vehicle list...
The old plate will be turned in to DMV and a new one issued along with a current registration. The computer records delete old plate and registration numbers after a certain period of time. Be glad, as Late registration fees can be 300% of normal registration fees (in addition to use tax, title and plate fees).
I've not tried it. But, you could bring the bare frame and engine to the CHP for number verification and HOT list search. If it comes up stolen, you only lose those parts, as they are the only ones "trackable". However, if someone feels like and investigation, don't be surprised if someone doesn't show up looking for the rest of the parts. Especially if parts start showing up on CL and eBay shortly thereafter. Also, if the Bill of sale indicates a complete bike rather than parts of same, better be prepared to answer questions about "chop shop" parts location.
I don't think most judges would accept "I was excited" as a good reason for you to keep parts you added to a stolen bike. Perhaps, "I was stupid" might get you some sympathy, but you better have dated paper receipts for any additions or purchases you made. You won't be reimbursed for paint or any rework you did to parts you can't legally own. (unless the rightful owner wishes to reimburse you, which probably won't happen if you are a hot head and show any animosity toward the rightful owner.)
Cheers,