I'm sorry - the doctors on this forum just answer an occational post now and then. An alternative is posting in the Heart Disease Expert forum (free, but limited to 2 questions/day) or the Heart Rhythm Expert forum ($20 fee, answered within few days).
I'm not sure if you consider me a "premium member" (I'm a community leader, not sure if that's what you meant), but I will try to give you an answer.
You are lucky to have a GP that is also a cardiologist. If a cardiologist is "really concerned" about your EKG, and wants you to take an echo, I would follow his advice. You ask about the cons with a TEE, as far as I know it's not that many, but did you get a TTE (normal echo) done first? TEE is most commonly used to investigate the atria, just like an esophagal EKG is used to investigate atrial electrical activity.
The procedure is usually done with mild sedation, and the risk of complications is low. The upside is that your upper heart is very clearly visible for the doctor, but as far as I know it's usually combined with a normal (standard) echo or done after a normal echo
It sounds like you have a lot of temporary blocks (you can't have RBBB and LBBB at the same time, this is dangerous and would have caused fainting, it's the same as a complete heart block). Maybe your blocks are heart rate dependent? I'm also afraid that your EKG findings and tachycardia are caused by your infections that are left untreated. Dental infections are known to possibly affect the heart (valves and other structures).
If your resting heart rate is in the 120-130s, (given that you are not extremely anxious every time you measure it) there is something going on that needs attention and treatment.
I can't do anything else than to recommend that you follow the advice from your GP/cardiologist..
This is a standard test they do before electrocardioversion, to rule out clots. It just gives them a better picture from the inside. I have had it, and wouldn't hesitate to have it again if indicated. Sounds like your doctor is trying to gather all the information he can to decide how best to treat you. Good luck.