I feel like this is something that I can accomplish with a Mikrotik, but I tried for several hours today with no luck.

eth1 - WAN, static public IP
eth2 - bridged to bridge1
eth3 - bridged to bridge1
eth4 - bridged to bridge1
eth5 - 192.168.31.250/24 (a different router is the gateway for this subnet/network) <-- I need access to this subnet from 192.168.1.0/24
bridge1 - 192.168.1.1/24 (DHCP enabled)

From any port on bridge1, I can access 192.168.31.250, but nothing else on that subnet.

I tried many firewall filters, various NATting, mangle rules... help!

Clean solution is to configure the other network to know about yours, so add route to yours to other network's gateway. Or just use srcnat on eth5, to make all connections to other network look like they come from 192.168.31.250:

Not sure if this is similar to your case, but this thread may give you some ideas......
viewtopic.php?p=891129#p891129