At least you can see all the DHCP options the DHCP server has passed you there. There is DHCP option for the domain it is on, and another option for a search list of domains.
Your router DHCP server probably only uses only a dozen or so, but the TLDR of the hundreds built in the protocol is: Option 15 for domain, and option 119 for domain search list. I know both work, as I’ve used them both via DHCP server.
I was just guessing at grep terms to filter through, as it included the ones I had. But I assume none of those options has your domain? (In the DHCP.OPTION[*] portion.)
I have DHCP4.OPTION[4]: domain_name = hosenet.local
Note: The OPTION[#] does not translate to the option numbers for DHCP, that is part of the protocol you can look up. These numbers, I assume, are just NetworkManager’s ordering, or in the order it got them.
If NM got either of those DHCP options, it should configure the search domain(s).