i dont know about "deep cleaning"
car wash soap, but maybe you might wanna give 1Z's soaps a try. says that it'll clean dirt and grime without harming your lsp, if thats what youre looking for.
for myself though if i want to start off fresh and make sure that theres nothing on the surface, i use PB bug squash concentrate as a pre-wash/rinse. you can also try using
P21S TAW. HTH