When I flew 737's the TREs would take great pains to point out the start EGT limits applied for the second start. I would always get them to clarify that if the first engine took, say 900° to start up, then I couldn't have the second engine back if it exceeded 725° during its start? (This was always preceded by an ash encounter, by the way). There is often an urge to invent arbitrary limitations based on something unrelated mentioned in the FCOM, I have found.
The way the checklist is written you have no choice, unless you want to go outside the QRH. It directs you to the Engine In-Flight Start checklist for the second engine start & that requires you to respect the EGT start limit of 725 degrees.