I've been looking at hiking trails and such in the general area.
Basically, I agree with
@no greed that it would not be possible to take a bike and a bike trailer from Mono Hot Springs across the Sierra to the Highway 395 area. There is a way to do this on foot only in summer, if you are truly fit enough and prepared enough as an experienced solo backcountry hiker with proper gear, probably including crampons and an ice ax depending on the route.
There are ways to traverse the Sierra from Mono Hot Springs on foot. The Kaiser Pass Road goes from Mono Hot Springs to Lake Thomas Edison. Vermillion Valley Resort is at the west end of the lake, where thru-hikers on the John Muir Trail (JMT), which is a segment of the famous Pacific Crest Trail, can pick up their next supply cache. This operates during the summer and there is a ferry that can take hikers from the eastern point of the lake to the resort and then back. Without the ferry operating, the trails around the lake to JMT are very difficult and have lots of elevation changes and steepness. Certainly not suitable for bicycle in April, without the ferry service.
Here is a description of Vermillion Valley Resort from their website:
"Deep in the beautiful Sierra Nevada mountains, the Vermilion Valley Resort (VVR) sits on the shores of Thomas Edison Lake, one of the largest lakes in the region. Just a short hike off the historic Pacific Crest and John Muir trails, VVR is a major resupply and layover spot for hikers who travel here from around the world to hike these internationally known trails.
Vermilion Valley Resort is one of the most remote locations one can drive to in the High Sierra. The VVR is located in the Sierra National Forest in the John Muir Wilderness. The last 27 miles of winding, often single lane mountain road adds to the allure and excitement as it is an adventure just to travel into the region by vehicle. The difficulty in reaching the VVR by vehicle has kept impact to the region at a minimum."
VVR is a hiker friendly wilderness resort located in the High Sierra Nevada mountains of California.
vvr.place
This trail joins up with JMT at approximately 7800 ft elevatoion and you have several routes to the eastern side, one perhaps to the north to exit at Lake Mary near Mammoth Lakes, the other to go east and south to exit at Rock Creek Lake near Tom's Place on Highway 395. Portions of these exit routes will exceed 10,000 ft elevation and are very isolated, without the number of hikers you would see on the JMT , so you are truly doing this alone in April and at high elevation where you need to be prepared for high elevation storms, snow, and high winds for days and nights on end.
This last segment leaving the JMT is the most technical and difficult of all. If you are not prepared as a hiker carrying all your own gear and with solid boots and winter-protective clothing, you are not going to make it. No way a bicycle can make it.
This gives you an idea of the route. The green dot is where one can access the JMT from Lake Thomas Edison. The north-south segment in blue is the JMT. The exit routes would be the ones in black heading to the north and to the east. The altitude profile on the bottom shows the JMT elevations, only, but gives you an idea of how high and rugged this area is.
Great campsites on Roaring Bear Creek, gorgeous lakes, and trails that branch to some of the best areas in the Sierras. Near Mammoth Lakes, California.
www.hikingproject.com