Don't know how to tell you this, so I'll just go for it.
Using two different colors of primer under a white finish coat will effect and show through the resulting finish.
You mentioned priming with white and the red putty shows through.
Light colors have trouble covering darker colors. Is that auto spot putty by chance?
What would have been the easiest why to get a nice one shade result,
would have been to use a "light" grey primer or silver. Then you could
give it a coat or two of white primer or flat prior to your finishing
coats.
For the light grey primer? I use the large cans meant for automotive
work. They are cheaper than the smaller cans with a modeling companies
name on it.
1.)
"WIll this help solve the problem?"
If you spray "everything".
2.)
"Will the light gray primer cause a difference in shade in the white that will be on top of it?"
Yes it will cause it to be just a bit darker, but not too bad.
3.)
"... it is completely in Japanese. How long should I let it dry before spraying on the white?"
Sorry, I don't know Japanese nor am I familar with Tamiya products.