When I got my ATP, I brought my civilian logbook (piston time,etc) and personal military logbook whose times matched my military flight records. I also knew exactly how many military sorties I'd logged (something like 800). Different airlines apply different conversion factors to military sorties for application purposes (some do .2, but SWA is .3 - https://swa.pilotcredentials.com/ind...qualifications). If you search the FARs or FAA website enough you might find definitive conversion guidance.

Anyways, conservatively 800 x .2 got me an extra 160 hours, putting me comfortably above 1500. My ATP examiner had no problem with this (I hadn't applied anywhere yet). Not really a need for a third "converted" logbook.