Trailer registration is managed by TxDMV, not TPWD. Key question is has the trailer been previously titled or just registered, or neither. Trailers only have to be titled if they are greater than 4000 pounds gross weight. Best thing to do is to track down the owner on record and have him/her sign the transfer documents, form 130-U. Regarding late fee costs, the 130-U form shows a 5% or 10% box if application is received after 20 working days of sale. Website shows significantly higher fees that continue to accrue with time. Better give them a call first.
I will add this - please don't sell your integrity and falsify the documentation. Not only is it illegal, it is also the wrong thing to do.