Ideas:
1) Monitor your front door. Footage from a front door monitor would have verified the in-and-out activities of the guest, and would have proven that your guests were lying about how long they stayed.
2) Take photos after every cleaning. It’d be unlikely that many roaches would suddenly appear in an area that was pristine hours before.
3) Ask Airbnb to ask the guest where they stayed the last few nights. This may not matter, though; I’ve read situations where the guest stayed the entire length of the reservation, didn’t say anything during the stay, and still received a full refund due to roaches.
4) Spell out in your House Rules that finding bugs does not constitute a reason for a refund. I have the following in my House Rules:
Guest also understands that Hawaii is located in a tropical climate, and that insects, rodents and lizards flourish in this environment. Although Owner will use its best efforts to hold to a minimum your interaction with these pests during your stay, Guest understands that any contact by Guest with such a pest within or outside the unit does not constitute a breach of this Agreement or give Guest any right to any refund or rental adjustment.
My House Rules also have the following paragraph:
Any issue that Guest has with the Unit shall be immediately reported to Manager via the phone number listed in the rental Unit’s directions and entry instructions. Guest understands that Manager and Owner shall have a reasonable amount of time to restore the Unit to acceptable conditions, and that any issues presented to Manager by Guest not preventing reasonable and habitable occupancy of the Unit shall not constitute a breach of this Agreement or give the Guest any right to any refund or rental adjustment.
This last point (#4) may prove to be the most protective - I’ve seen several cases in which Airbnb upheld a particular host policy because it was explicitly spelled out in the host’s House Rules.