It sounds like the issue you're running into is 2 parts:

1. Using a dedicated device for Internet connectivity.
2. Broadcasting wifi evenly throughout the house.

I think the best implementation that stays within your constraints would be to purchase a hotspot with Ethernet capabilities (like MiFi or Cradlepoint) and place it where you can best get reception. Then buy a couple meshing access points like Ubiquiti APs and place them throughout the house. Run an Ethernet cable from the hotspot to one AP and then mesh the rest. If you can run Ethernet cable to each access point using a network switch, that's even better.

A cellular signal isn't one single thing. A GSM booster will improve the availability of a GSM signal, but that's calls and SMS but isn't going to do a lot to provide mobile data given that 3G coverage is basically dead and 4/5G data doesn't operate on a frequency that the GSM booster will be doing anything for. You can get cellular WiFi hotspots that you put a SIM in and they basically become your WiFi router or just use a cellphone that has hotspot functionality

My advice for this type of situation has been a 4G/5G modem-router+directional external antennae.

They are more sensitive so they can work with a weaker signal, and they don't move once installed somewhere, pointed at a cell tower. That means the connection can be made way more stable than when using cellular data on a phone.

Combine that with a prepaid data SIM, and you wont waste any money when you're not there using it.

I'm not sure about a GSM booster, but a smartphone can be made to emit Wi-Fi, whose range can be extended with a Wi-Fi Extender. Place the phone into an area with good coverage (by a window perhaps?), activate the WiFi hotspot, and connect your other devices to the phone's WiFi network. If the range is insufficient try switching to a lower band (2.4GHz signal will travel further than 5GHz) and if that's still not enough buy a WiFi Extender, set it up to connect to your phone's WiFi network and connect your devices to the Extender's network.

Check out WeBoost brand repeaters. I live in what used to be a rural area and when we moved in the cell signal was trash inside the house but fine outside. Put an antenna outside on the roof, ran some coax cable to the kitchen and mounted a repeater there. No issues. Works for all major cell bands.