Liquid Web can actually help in this area. They do not offer shared hosting packages any more but do have the Virtual Private Servers available, packages of which do give you SSH access. I currently use them and have had no issues whatsoever. I currently have two Flarum installations where I did use Composer so I know it works without any hassles.
Their support is second to none as well which is something you get what you pay for with.
If you decide to go with them, would you mind using my referral link? If you want to go for a Storm Dedicated Server I can provide you with a one-off $80 coupon as well.
I was planning to release a free hosting service for FluxBB and Flarum forums but I have put that on the back burner for now.
