OSHA actually addressed logos for flame resistant clothing (arc rated clothing) related to arc flash in an interpretation letter. This is interesting. They didn’t give specific guidance but indicated the employer is responsible.
ASTM F1506 does not require logos to be flame resistant but in a non-mandatory Appendix X1.2.5 it says “Logos, name tags, … are used to identify …If these items are constructed of non-flame resistant materials (such as polyester or rayon), their overall area should be minimized … the use of several logos over the entire garment should be avoided.”
For arc flash issues some use a non-melting thread like Nomex, Kermel or Conex. Other options are coming online all the time. I’d bet a Protex modacrylic thread is available. Colors used to the the big limitation and non-melting threads were often more brittle. This is much easier to work with today with new thread technologies. Atlantic Thread is a supplier of these in the US and others are out there.
Non-FR embroidery works as long as the size is small. The burns I have seen are not from embroidery ignition but from sweat build up UNDER the embroidery which can happen with any embroidery. One manufacturer recommends nothing over a business card size and that seems like a good limit but the standards like ASTM F1506 do not put this limitation on the logo.
If cost is an issue, you will probably see that it isn’t much more than a dollar more for non-melting thread. The ones who charge more don’t usually do it much.