For any interested:

I asked the physio who is treating me for his opinion on accepting the scalene block injections, and shared my concern about combining treatments.

His opinion was rather different. The points he made were:

# You, me, your GP, and Neurologist don't know the cause of your TOS
# We are treating your symptoms now, and if that provides relief, thats great
# It is not necessary for you to try and act as a single subject controlled study
# So you should go for the injections if you want to try them
# If you get relief after taking the injections, thats great, it is not necessary to identify causation between the physio and the injections