DHCP is an element of PXE. There is no PXE without DHCP.
There are older methods of network booting, such as DHCP's direct ancestors BOOTP and RARP, but then, those are also unrelated to PXE other than the shared ancestry.
I'd say the simplest way to do what you want is to run a DHCP server on the "other working PC on the local subnet". You likely already have a DHCP server, but if it's just the one within your Internet modem/router/gateway, it's probably not very featureful. You'll thus get other benefits by replacing it.
The two standard options are ISC DHCP and dnsmasq. Most of the instructions you'll find on PXE assume ISC DHCP, and it's fairly easy to set up, but
dnsmasq is easier in some ways and also gets you a DNS server suitable for managing a small LAN.