It would be worth asking at any of your universities that teach agriculture, soil science, or forestry to ask if they have a soil science lab, and do they test samples from the general public. Our local university doesn't test for free, but they can test at low cost. I hope some of your universities do the same.The £33 fee from RHS does seem high to me.
When RHS says multiple samples, they probably mean multiple SUB-samples (often from 8 or more locations in an area you want to test). The sub-samples are all placed in a single bucket, mixed thoroughly, big roots or rocks pulled out, then mixed again; this is a composite sample. What you send to a lab is a portion of the composite sample, you pay only for that one sample. The lab usually only needs a cup or two of soil to do their testing. The reason for doing a composite is that soil is very heterogeneous, a composite sample can be more representative than a single sample taken from one location.
I've attached a photo of my soil sampling kit. You can buy a well-made stainless steel soil probe, but they are expensive; mine is made from a scrap of 1-1/4 in PVC pipe, plus a T-connector for the pipe. A plastic bowl or bucket and a plastic rod to push soil out of the probe (and a cheap stainless steel kitchen spoon, not shown) completes the kit. Fine for collecting soil from ground that has been worked a few times recently. Inexpensive and easy to use. I may post information on making it and using it later on.