Hi,
Assuming your listing was sound and accurate, it is not a good idea to accept their reason as you will lose your insertion and FVF fees, making a second chance offer will incur a second dose of FVF on that item.
Reconsider your position, normally I would <edited to add> "(if I intended to make a second chance offer)":-
1) File an unpaid on the original winner.
2) Once that has completed and your fees have been reimbursed then make the 2nd chance offer.
Two problems with this, one the winner may neg you, but this can probably be appealed against now, in the old days they would risk a red and were unlikely to do so.
Secondly I rarely found that after a week+ it could take to complete an unpaid the heat of the moment is lost and the eligible under bidder would rather wait for you to re list and get it for less.
If they do not take up the offer you have lost your chance of a free relist, so ask them by ebays message system to enquire if they would be interested or not, if no reply within a few days just relist.
Making a second chance offer whilst an unpaid is dispute is in progress can cause feedback problems as the system gets confused, however that may not be the case now.
Sorry it is a bit involved, personally I would just raise the unpaid follow it through and relist from scratch and say in your listing that this item is relisted due to a non-paying winner. In that way new bidders have are better informed about why it is being relisted, but do not specify the defaulters name or ID because they can get eBay to pull your listing and you may get penalised by their listing police.