As jaymor says, you really need legal advice. The problem is that English law (I assume your FiL is in England or Wales) only really pays out liquidated damages if you can prove an economic loss (I know that is a gross over-simplification) and then only if the care level amounts to a breach of contract. In any case, you would probably at most only get a sum equivalent to part re-imbursement of care for the period during which the sub-standard care was alleged to have taken place.
It's worth putting some speculative numbers on it: if you have a high-end care home (£60kpa = £5k pcm), and - say 70% of that is assessed as "hotel costs" (accommodation, food, depreciation) and "staff costs" (the required level of staff numbers) then any reimbursement would be probably amount to a percentage of the remaining £1500 pcm - say 20% x 4 months =£1200. That's high-end - if the costs are lower, the likely payout will be lower. (English Courts don't usually do punitive damages in this sort of case, btw.)
It's worth contacting a solicitor and using one in this instance as they will have a more accurate idea of what are realistic numbers and likelihood of success. It may be that the costs of litigation (especially if the sum likely to be paid out is below the Small Claims Court threshold) exceed any likely payment, aside from the chances of winning. Also it's evidence of seriousness. Claims for reimbursement for alleged sub-standard care can be a pretty regular occurence in care homes and you have received what is probably the standard response. Using a solicitor pushes your case above the "background noise".
And the bit about the NHS and CQC not being able to enforce reimbursement is entirely correct. As your FiL is self-funding the NHS is totally irrelevant, and AIUI the CQC has (currently) no powers to order compensation.
W