I would definitely consider using a lye based product.
If you do, and it doesn't work, be certain to give it time to neutralize itself before you open up the plumbing.
About the plumbing, do you have a basement, crawlspace or slab foundation?
What floor is the bathroom on?
Is the bathroom sink backed up or is it only the tub and toilet?
If there is a basement, is anything backing up down there?
The answers to these questions can help you know where to insert your auger or snake(or hand! Gloved of course...), and what kind of tool will work best.
A hose or bucket of water can help you know for certain when the drain is open.
Even after it seems to drain freely, test it with a bunch of TP.
Even something as simple as dental floss can cling to the sides of a pipe and catch debris.
This is why the rule is"#1, #2 and TP are the only things you should put in a toilet bowl"
An auger can sometimes punch a hole in the debris and the water can flow through,while the original obstruction can remain.
Only by putting more debris down the toilet,and testing for function, can you be certain the job is done correctly.
I usually flush 5 heavy loads of TP down any toilet I've augered.
If it can take that without backing up again, its fixed.
Drain pipe replacement, when necessary, is generally a dirty filthy job, but not actually difficult.
Good luck, keep us posted!