The correct color for mk1/mk1a wheels was "foam white", not moonstone. (but they are pretty close in shade!) MK2 wheels were silver (dull). The backside of the rim was always black. The black was over-sprayed over the foam white, or silver. It was pretty sloppily done. As for tires... The closest radial to 5.90/13 would be a 165 or 175/13. with a "80" aspect ratio. Doubt you will be able to find a 175 (80)/13 easily. 70 series radials will be (obviously) smaller in overall dia.
The Healey guys run into the same issue... a 165/80/15 looks quite a bit smaller than an original 590-15. A 175/80/15 fills the wheelarch better and look more "correct", but they are hard to find and very spendy.. Dunlop used to make bias ply ties in both the 5.90/15 and 5.90/13 sizes, but has discontinued both... Sprites used the smaller 13" tire too.
I had no problems sourcing a 165 (80)/13 tire here locally.... (Altho it was a Chinese off brand)
Not like trying to find 5.25/5.50 x 16 tires for my father's Sunbeam Talbot 90 MK2a DHC!! I kept on getting asked if I wanted a tractor tire!