Toilet plugging issue, need some advice on options
Hoping to get some advice from folks on here. So we have a basement bathroom with a toilet, stand up shower and sink. The house is from the 1920's and this bathroom was probably added within the last 10 years prior to when we moved in. We have been in the house for almost 6 years and have never had any problems with the toilet or anything else in this bathroom. Within the past few months the toilet has started acting up. Sometimes it flushes normally. Sometimes the water fills up and then drains very slowly. Usually when that happens the water goes all the way down to the point where it is so low there is "burping" of the water, i.e. i'm assuming this is air releasing. So this morning I had a reputable plumbing company come and look at it. They think either there may be blockage somewhere or they plumbing may not have been installed properly when the bathroom was built.
So the options right now are: a.) $400 to pull up the toilet and they bring a machine in and do a snake. No guarantee this will work. b.) run a camera down to see if they can tell what the problem is, this is $250. The other option is that the plumbing is set up wrong and they would have to re-do it. I'm sure this would be very expensive and it would be a last resort for us. The guy mentioned it was possible the plumbing was set up incorrectly and there was no venting or something like that. We have never had a problem with the sink or shower water pressure or draining. We also have a washer and dryer hooked up in the basement along with a big sink beside them and have never had a problem with any of those either. The toilet has worked fine for years up until a few months ago. I'm assuming if we had plumbing issues or no vent that this would have appeared early on after we bought the house. The only thing I have done down there recently is turned off the valves leading to the outside spigot so the water wouldn't freeze per my neighbors advice. That was a few months ago. The valves are located in the ceiling above the toilet. I did buy a snake a few days ago at the local hardware and tried to snake the toilet. It was roughly a 4 ft snake. It seemed to go down as it should but it didn't do anything.
So basically I'm looking for some advice on what to do here. I didn't expect the problem to be so costly but at the same time I need to get it fixed. I would appreciate any advice on this issue to help me make a decision on what my next step should be.