four way switch to three way for vent fan
You want a DPST timer switch (or a relay) since your existing switch is a double pole switch, not a 4-way switch
From the wiring arrangement shown, the only way things could work is if your existing fan/HRV combo switch is a double pole switch, not a 4-way switch. As a result of this, we need two switching poles at the living room location, and most timers that support that are made in a larger &quot;time switch&quot; form factor that's designed to be surface mounted in an unfinished space, such as a mechanical room or garage.
There are a couple options here, but you'll likely have to replace the box with a bigger box, either way
Fortunately, you aren't out of options, as there are a few wallbox timers available that do provide double-pole switching, and you can also use a relay to provide the second switching pole due to the fact neutral's available at the living room box.  Most of these, though, will require replacing the existing single-gang box with a two-gang box in order to make sufficient room for the parts required.
Going old school here
The first option I'd look at if the box replacement is practical is a Marktime 72131AB timer.  This is a two-gang, spring-wound, DPST timer switch with a hold function that lets you override the timer for both poles; the black wire and jumper go to L1/L2 on this switch, with the black furnace wire and the red fan wire going to T1 and T2 (or vice versa), and no need for a neutral wire.
If you want a fancier timer...
If you want a fancy electronic timer instead of an old-school wind-up timer, that's possible too, but requires a timer that uses neutral for its power return, such as a Lutron MA-T51MN, and a 120V SPST or SPDT relay such as the AP&amp;C PAM-1, in addition to likely needing extra box space to fit it all in for practical reasons. (The extra box gang is covered by a blank faceplate.)  In this case:

A white pigtail from the silver screw on the timer and the white wire from the relay module get nutted in with the existing bundle of white wires in the box
The green ground screw on the timer is pigtailed to the existing ground bundle in the box with a green or bare wire
The incoming black hot from the bathroom is nutted to the blue wire on the relay and a black pigtail to the black screw on the timer
The outgoing black switched-hot headed for the furnace is nutted to the black wire on the relay and a pigtail to the brass screw on the timer
The outgoing red switched-hot headed back to the bathroom gets nutted to the orange wire on the relay
And the yellow wire on the relay is capped off by itself

If replacing the box isn't an option
If swapping junction boxes is a non-starter, it's still possible to do this, although you have to give up the notion of a &quot;hold&quot; function on your timer to do so, as the Intermatic FD460MW (aka the only DPST timer I could find that fit your existing box and faceplate) is not available with that feature.  If that is acceptable, though, you can wire said timer as the existing double-pole switch was wired, with the black hot wire and jumper to the LINE terminals, and the switched-hot wires (black and red) to the LOAD terminals on the timer.