Note: I am not from North America, have never been to Mt Rainier, and have no experience on the terrain or local conditions at any time of year in the vicinity. So, take advice with a grain of salt and consult on-the-ground experts like your guiding service.
I had a look at the guide for the Disappointment-Cleaver Route Brief(PDF), from the National Park Service page on climbing Mt Rainier,
From this I think I can say a definitive No. There is extensive snow travel, and a fairly complex route, that if it goes wrong, will lead you into difficult terrain with crevasses, melt-holes etc. To quote pg 12 of the guide:
Camp Muir and the Muir Snowfield are nearly surrounded by
glaciers: the Nisqually Glacier to the west, the Cowlitz
Glacier to the north and east, and the Paradise Glacier
to the south and east. A minor error in navigation may
lead you onto these glaciers where there are numerous
crevasses and other hazards.
To deal with these you would want a boot that can handle crampons effectively. Lowa Renegades appear to only be suitable for light strap-on crampons, not proper crampons.