Ok, apparently the nameplate has been removed, as every one comes with the specifications on the nameplate with the model number. That is standard for UL.
If you cannot get a new circuit out of the panel for the treadmill, your only choice is to get a transformer and install it on a cord so that it can plug into the dryer receptacle when you wish to use the treadmill.
You cannot wire it direct, as the dryer circuit cannot be shared by code, it must remain independent.
Then you would plug the transformer in when needed.
A 2400 watt transformer will work fine, that is the high end of the circuitry so it will cover with no problem.
The concern I have, is the unit was running fine for a while and only now exhibited the issue of tripping the breaker INSTANTLY, which is normally a problem inside the treadmill itself.
I think the treadmill has a bad motor or controls at this time from the information.
Do a test and plug the treadmill into the washing machine receptacle, it should carry it with no problem. The washer is a 20 amp circuit by code.
Kitchen receptacles are also 20 amp circuits and could be used as a test.
I think you need to verify the problem is the treadmill has failed or the house circuit, before spending for transformer and connections that may not work either.