I'm not sure I agree with this - they were specificially chosen because £50 a month for the spec of machine and backup of the entire site and its content to another country (in case the datacenter burnt down) was a total steal at that price. They've since bought their prices inline with most of their competitors but gave a years grace to their existing customers so this increase should not have been news to you or any of the administrators.
You might want to consider having Invision themselves host the site (currently $89/mo) - I can't tell if that offering does everything you currently have but you could ask them - this might also include the software license cost. I don't think you'll be able to get this site to run on a cheapo (£5.99/mo) webhost designed for peoples unread blogs or local interest websites written in Word '97 because they don't offer you full access to the underlying VM or have limits that the size of AS exceeds.
All the smaller cloud hosts (Linode, Hetzner, OVH, whoever), as you currently use, are priced about the same. The larger ones (AWS, Azure) are all more expensive.