I believe Ryan and Prashun above hit the nail with the vent stack--
I pulled the old toilet, dumped a bucket of water down the drain as fast as I could and it took it all no problem, so I put a new toilet over the hole--
And it did the same thing as the old toilet...
After removing the toilet I aimed the water line into the drain and running wide open, the drain took all the water easily...
Then I found a plunger... And the instant I pressed down over the drain pipe, I got back pressure. I could push the standing water down the pipe, but once I removed the plunger, all the water came right back. The one time I plunged about 10 times, I had to mop up water that blew back out of the drain. FWIW, the standing water is at least 2' below the drain opening. And when I plunge, I can hear water moving thru the pipes under the floor towards the bathtub, but the sound didn't seem to come from the tub drain. And plunging the tub drain, there was no back pressure at all.
Not looking forward to crawling under the mobile home in 20° weather, and not exactly sure how to troubleshoot and repair a vent pipe?
And since this is a mobile home and we pay an exorbitant amount of lot rent where it's at, I have one more question: Who's responsible for fixing the sewer line, me, or the landlords?