Boy this is a tough decision... and impossible for anyone to answer without having seen your images and doing an exam. I understand both perspectives... if you are hard on shoulders as it is and are you going to end up tearing your damaged labrum anyway... thus indicating a more aggressive approach? Tough to answer.
In terms of the biceps: tThe insertion of the biceps is two fold (hence the "bi" prefix) have a long head inserting in the shoulder joint and the short head inserting on the corcoid process of scapula (http://healthtwo.wikispaces.com/file...ps-brachii.gif
). Doing a long head tenotomy or tenodesis does not have strong indications other than pain relief. At this time far they are unclear of the exact function of the long head of the biceps. Thus it has been subjectively found in pt's with pain despite initial arthroscopy that if there is damage to biceps tendon transecting it will provide symptom relief.... and only minor functional deficits. Thus if one's pain goes away and function is reasonable... patients are happy. In younger pt's (<50 years) tenodesis is indicated but in theory only... the idea is that you still get function benefits of long head of biceps by tacking it down to the bone. However there are few good studies that tenodesis has better results functionally or pain wise than tenotomy.
Overall, it's probably going to be a gut decision for you. Thoughts to have is: How well do you feel each of these fella's understand the demands you are putting on you shoulders? How well did you feel like what their assessment, recommendations, and discussion match with how your shoulder feels? How closely does the surgeon's thinking and philosophy mirror your own?
Tough decision in the least, but I think overall either way you will find a benefit!