This "Hard Drives: 2 x 1TB NVMe" suggests the machine is using fast SSD drives.
That's the problem: you could get a vast amount more storage on hard drives, but they're much slower access than SSDs.
Or, you can pay quite a substantial amount for large NVME SSD drives.
Maybe FH will be willing to buy and install larger drives than that, if you pay them the drive cost price plus a bit for their trouble?
Can't know for sure the drives below are compatible, but they're probably the sort of thing you'd need a pair of (identical pair is best for RAID 1)
https://www.cclonline.com/product/3...-2280-PCIe-Gen3-x4-NVMe-Internal-SSD/SSD1343/
https://www.cclonline.com/product/3...3-x4-NVMe-Internal-Solid-State-Drive/SSD1320/
So you're looking at £900-£1,100 total for the 2x 4TB NVME drives, plus whatever your hosting company would charge you to fit them and copy the data across.
You may have to haggle with them hard to let you get the drives as a one-time fee rather than see them add a ridiculous extra amount every month but they lease the drives to you.
If you go with my suggestion, that instantly nearly quadruples your storage space on the server.
Alternatively, get the hosting company to add a cheap and cheerful hard drive, and arrange to get stuff like the web server log files to be stored on there, saving the main SSD for your site's data.