No, you do not have to cancel the contract to ask the seller/lender to pay for termite work. If you cancel the contract, there is no guarantee they will accept a new offer from you.
And, you are correct, lenders will seldom cover repairs on a short sale but it's not completely unheard of. If you have not removed all your contingencies yet (which hopefully you haven't yet), ask your Realtor to possibly do an Addendum to the offer to cover termite work - based on the Termite Inspection. There is no guarantee the seller/lender will agree to the Addendum but for $5000, I think it's worth at least trying. If the seller/lender do not agree to the Addendum, then you need to decide how badly you want this particular house (enough to pay an additional $5000?). The seller/lender may agree to some, if not all, of the repairs if they want to get this house sold - especially since it's been Pending 3 prior times already.
On a couple of Short Sales I've done, the Buyers Agent and myself have contributed to the repair costs (from our commissions) if the seller/lender won't - to save the deal.