You have my sympathy on the basement flooding issues. Been there. Thankfully not sewage backup in my case.

it’s definitely not an issue you want to let go for long. If there’s an obstruction, it will get worse quickly as more solids go down the drain. Easiest way is to just hire a plumber. Complexity and cost are going to depend heavily on the cause and location of the obstruction.

If you want to try to DIY it, you could try to snake the drain but on a 3 or 4" line you’d need a good sized power auger to make any serious headway.

If it were me, I would get a sewer inspection camera. Low end ones start at less than $150. This is really the only way you’re going to find out what the root issue (no pun intended) is outside of paying a plumber to come do the same thing. A large wet dry vac also comes in really handy in these situations.

How old is your home and what’s the sewer line made of?