I'm afraid that I don't get it. You mention "don't want to let your friends down".
What are their expectations of you based on? Why on earth did they select and pay for this particular 'Pro' if they don't have the confidence in them that they will get the shots they need? So much so that they feel they need you there covering the wedding because your shots will be so much better (after all you are Cartier Bresson reincarnated to them)? At least I hope the bride doesn't have more confidence in you than their pro. I hope you get my point. Don't put the extra pressure on yourself as to how good you must be or how important you are to getting the shots.
On the other hand, it may be that she just wants someone that she has invited who is competent photographically to take photos throughout the day - that compliment those photos she will get from the pro - or maybe shots they aren't likely to take or times of day they may not be covering the event. I can see the importance of that and it may already have been discussed and agreeable to the pro.
It concerns me when you are talking multi-light setups and having a second shooter with your wife etc? I'd have to ask then - why is she hiring another photographer? I would presume that you may end up with conflicts with the hired pro and you may be jeprodising the effectiveness of their resulting coverage and images because of the irritations that that would probably cause.
Most significantly is the fact that there are very few 'pro' photographers who do not write concerns related to additional photographers into their contracts. Some state that no other people can take photos - while more common is a tolerance of photographers as long as they stay out of the way of the rightful working photographer. My contract allows people to shot as long as they don't interfere with me - - - but also no other photographers are allowed while I am photographing my signature shots (the time I have alotted for formals). And never presume that the bride fully perceives the importance of that agreement, even if she has been made aware and signed to it. If the photographer walks off the job or refused to do certain things because you or the bride have violated this agreement, you may have all the pressure on you to perform the rest of the photography LOL.
My advise! Nothing wrong with introducing yourself to the photographer if you feel like it and you are not interfering with their work. No need to go into details by stating that you will not be making money from your photos. The photographer doesn't care. Just stay out of their way and if they bark at you because you are an irritation - back off. I've had guys come up to me at weddings I am shooting - holding Canon or Nikon pro bodies with massive 85mm f1.4 or 70-200 f2.8 lenses trying to impress me. While I am courteous and may say nice camera etc, I am there working and not interested in what gear anyone has. If they get in my way, I don't care how impressive their gear or how important they think they are to the bride or groom - they will be informed of my desire to have them refrain from interfering. I think that would be pretty standard with most pro photographers.
So act like any of the other guests invited to the wedding, and take your photos from the outside without interfering with the paid photographer. Because you have more knowledge and better equipment than some of the guests, you should be able to find angles and make lens choices that get you some excellent shots acting in that role. Maybe ask the bride if there are any parts of the day that the photographer is't providing coverage - and if so, you can then step up to the plate and provide the extended coverage so the whole day is documented.
As far as equipment and gear. That is the easy part. A camera and lens are all that's needed. Doesn't matter which ones. If you need to use flash, keep it on the camera and bounce it off of a ceiling or wall. You may be surprised that it is a challenge to shoot with flash outdoors in bright sunlight - because flash sync speed restricts the shutter speeds that can be used.
HERE IS SOME VALUABLE HELP ON FLASH LIGHTING FOR WEDDINGS:
http://neilvn.com/tangents/bounce-flash-wedding-receptions/
And finally - - - it could be that your real desire is to learn and practice taking photos at a wedding so that you can move into being hired for a wedding. Nothing wrong with that. I did it on a friends wedding 35 years ago where a professional photographer was hired. Many pros probably got their start that way. Photographing your friends wedding for that purpose - all that I have stated above, applies.