I agree with Alex, Liberty Ridge and Ptarmigan Ridges are typically in sub optimal condition by early July most years.
Perhaps the Mowich Face might be a good choice for that time period. Probably icy, but also might be less exposed to loose rock than LB or PR.
Do you have your heart set on Rainier? Mt Baker is, in my opinion, more enjoyable and very similar to Mt Rainier. More of a wilderness experience, no entrance or climbing fees, less red tape.
North Ridge, Coleman Headwall, and Park Glacier Headwall would all be steep, interesting, and should still be in good shape in July.
Dispensing piss poor advice for since October, 2000.