EDIT: I’ve attached a rough map of the situation. The laundry room is the little room in the middle. The red dot is where the dryer vents into the garage....
Could you replace your existing dryer with a ventless dryer? Then you wouldn’t have to worry about running ventilation - you would instead periodically empty a container of water from the dryer. I’ve been using a ventless dryer for a year or so (my house also has limited options for proper ventilation), and it’s been great so far.
Ideas for electric clothes dryer that vents into attached garage
EDIT: I’ve attached a rough map of the situation. The laundry room is the little room in the middle. The red dot is where the dryer vents into the garage....