Grow room space usage

If it is questionable whether or not I will need a filter for the veg room, would it be safe bet to just install a 4" fan w filter?

Another thought, could I exhaust air from the veg room into the flower room, then exhaust it through a filter to the outside from the flower room? This would also supply warm air to the flower room. Or will the air from the veg room be too oxygen rich?

That's a judgement call you'll have to make when you start growing. If they start stinking, throw it up. You're going to want one for the flier room anyway, so you can always buy the second one later if needed.

As for venting into your flower room, if you're exhausting it there shouldn't be much issue. Just spend some time with them and they'll get all the CO2 they need. Or just run a CO2 setup if you're worried about it.

Filtered & vented exhaust does more than just kill the smell...which it DOES in veg; not as stanky as bloom phase but mj plants can be pungent even in veg phase once they get big enough. Seedlings hardly smell at all but 4 fully vegged plants will stink up your house if not filtered or at least vented to the outside.
Yes I double filter my rooms as well. Sure can even connect to a single filter if you get a Y split connector for your ducting. I filter the air going in and out & there's almost no smell at all. You can use the air from one room to feed the other if that will work for ya.
Using T5s in your veg area won't throw a lot of heat but I assume you plan to use hid lights or even LEDs in your bloom room the exhaust can help dial in temps especially if you add a speed controller. I suggest getting an exhaust fan that will exchange the air very quickly & give negative pressure so your plants will get good supply of fresh air, your room temp/humidity will be optimal, & minimal stank. This is what a lot of noob growers cheap out on & end up having issues down the road.