The clay you really want is fireclay, that company is very, very expensive £35.23 for 25kilos plus £15 delivery!!! but I understand your local.
I used
www.potclays.co.uk same clay £12.76 plus about £10 for delivery. Arrived in a couple of days.
Any clay you buy is 1 part of the homebrew mix, if you buy pizza clay?? you'll still need sand, Portland and lime.
Why don't you consider casting your dome using homebrew, it's the cheapest option.
I had to build mine on a budget so that's what I did, for my 33 inch oven, I splashed out on firebricks for the floor, but you've got storage bricks.
I also had to buy blocks, but you've got that covered as well.
I am doing other work so I made a large order from BNQ, free delivery over £100 with a trade card. The cost of the materials to cast the dome were, bulk bag of sharp sand (but used nowhere near the full bag but it is part of another project) £40, bag of Portland £4, bag of lime £12, clay as above, £22. I also put polypropylene fibres in the mix that melt at 160c and aid the curing fires. £10
So the actual cast dome cost less than £90, with the exception of the fibres (but you may want them in your final render coat for strength) you'll have to spend this anyway if you continue with your original plan.
You, have nowhere near enough insulation, I managed to buy enough body soluble ceramic blanket for a 3 inch layer from ebay for £66.
I was fortunate enough to be gifted 120 kilos of pumice by a member on another site, although I spent days crushing it!, I used every speck of it. I used thermalites under the floor with 50mm of pumice on top of that.
Once the ceramic blanket was on I put about 60mm of pumice over that.
The whole process was very simple.
You make a sand mould and a entrance gallery from timber, cover the sand in newspaper, that took me half a day.
cast the dome, did it the next day, half a day, cover the dome in a wet sheet for 7 days, but take the sand out after 2 or 3.
After 7 days take the sheet off, which you've kept damp, leave it another 7 days, weather dependent, to dry
a bit more. I installed the flue whilst waiting, and I did have lovely weather.
Blanket went on and 10/1 pumice/ Portland over the top, 3/4 quarters of a day.
Curing fires 2 days later.
Still got to do the finish render coats.
Where are you in Kent?