I think I know what the problem might be here.
As the club don't have wage budget remaining I can't offer the part time wage that the player wants. However if I switch the contract to full time I have to pay the league minimum wage of £425 p/w at least, so I can then sign the player. I shouldn't really be able to sign full time players at a part time club and it's a bit of an exploit as it seems I can sign players indefinitely on £425 a week even though I have no budget left.