Another vote for north in July. I count Yosemite as the end of the northern segment, after that you have the big drops into the Valley and big climbs back up into higher elevations, plus the desert segments.
Also be flexible, if one pass is closed due to snow, usually a lower one to the north will be open. Lassen Park Rd was closed into July in 2011, but there was a (less interesting) route around the mountain.
By the way, a climb up to the Lassen Peak summit is quite interesting. It's a 2000' climb in 2.5 miles, and I negotiated it in biking sandals. When I did it a few years ago, they had a temporary ranger station and store in the parking lot, so I felt reasonably safe locking my bike there, but it may since have been removed (Google Earth's 2010 image shows it gone) You may just have to hope nobody goes through your stuff while you're gone. If you do the Peak hike, I recommend camping at Summit Lake the night before, then after the hike, it's a big downhill to 5000', followed by a mostly flat ride to Lake Almanor, usually with a pretty good tailwind.
A shorter hike is to the Bumpass Hell geothermal area, the trailhead is just past the summit.