How far is your outbuilding from the nearest point of the house? If you don't want to run ethernet underground you could get a few cheap mesh nodes. One beside the router, one at the window nearest the outbuilding and one in the outbuilding.
I just done something similar for my uncle for 70ish quid, he wanted a quick cheap solution before something more permanent but has ended up keeping it. As long as your not expecting heavy traffic through the network in the outbuilding it should be fine. He uses his to play TV through a firestick as well so similar situation to yours by the sounds of it.
Wired underground is definitely a better option though if there's trenches being dug anyway. A cheap Asus router can be set to use as WAP without flashing new firmware.