I am not sure I understand the question, so I may be overly "elementary" here. If so, apologies.
The fresh air kit (or outside air kit) for the stove, pipes in air from outside directly into your stove. That does not really affect the room air quality either way as it's a (mostly) closed circuit from outside, to your stove, into the chimney to outside - unless your outside air is "dirty" in which case avoiding having air leak in is a good idea (which might only be the case in wildfire season, where you live, I surmise). This is because the stove needs air to burn, and without an outside air kit, it's using room air. I.e. it's sucking air from your room, uses it to burn the wood, and dumps it into the chimney. All due to the draft, i.e. low pressure in the chimney. The implication is that new air has to leak into the room/home.
For tight homes, air leaking in is minimized, and that can lead to problems burning the stove (smoke roll out etc).
An outside air kit avoids this. That is its main goal. It is not meant to keep the air in the home at a higher quality level (other than indirectly by avoiding smoke roll out for too-tight homes).
The HRV is meant to refresh air in the home, i.e. to keep the air quality in the home better, and will (should) not affect the stove burning negatively (in fact, the HRV will avoid negative pressures in the home/room, which might help avoid smoke roll-out issues).
So yes, if you install the stove, with or without the outside air kit, I would put the HRV in regardless if your home is tight: if you don't have the outside air kit, you need the make-up air leaking in thru the HRV. If you do have the outside air kit, you still want to maintain a fair air quality in the tight home.
And yes, if you have the opportunity to move the HRV to another room, e.g. kitchen, that would be a good idea, imo.