You could have a plumber come out and clean the line and locate the issue for probably a reasonable price. Once the issue is located, you could dig the pipe out yourself and then have a plumber repair or replace it to save money.

I watch a YouTube channel called Drain Addict and this dude uses a pressure washer with special heads on it to clean out drains and cut away roots. He then uses an inspection camera with a locating beacon to identify where the line is buried, where the break is at, and how deep.

Can always rent an electric 3/4" snake from a box store for under $50 I think. Make sure you're close to the feed point or you'll be wrestling with sewage-soaked coil. I'd go from the floor drain that is backing up if you don't have a toilet you can pull right next to it. Goal is to feed it to the sewer main. Scoping is the right answer, but a powered snake rental and an hour of dirty work has a chance of giving you some breathing room.