I found the solution. Maybe it will help someone:
HAProxy is intended only for reverse proxy behaviors, so don't bother with that. Their website even explicitly says so, and recommends using Squid if you need a regular proxy server.
So, install the Squid package in pfSense, click Save on the Local Cache tab, adjust the settings on the General tab and click Save. In my case, I needed to set the listening port on the General tab to some random sacrificial port number that I'll never use, because the real ports will be handled by my code below. Now, scroll to the bottom of the General tab, expand the Advanced area, and put exactly this text in your box labelled "Custom Options (Before Auth)", except replace the IP addresses and port numbers with your own:
##### START MY CODE #####
# these are the IP addresses and ports that Squid should listen on.
# Remember, ignore the listening port you put in the pfSense GUI.
# I did not actually need these loopback IPs in my code but I included them here in case it helps the copy/pasters.
http_port 10.200.1.1:8001
http_port 127.0.0.1:8001
http_port 10.200.1.1:8002
http_port 127.0.0.1:8002
http_port 10.200.1.1:8003
http_port 127.0.0.1:8003
# Give each port a name
acl portA localport 8001
acl portB localport 8002
acl portC localport 8003
# Map each port name to the IP address you want the traffic to use as it leaves Squid. These IPs must match the WAN IPs assigned to your pfSense interfaces and/or Virtual IPs. Squid will automatically put the packets onto whichever interface these IPs are assigned to.
tcp_outgoing_address 111.111.111.001 portA
tcp_outgoing_address 111.111.111.002 portB
tcp_outgoing_address 222.222.222.001 portC
##### END MY CODE #####
