The answer to your first scenario is, probably not. You should ask the flight club though. They may make you sign a contract saying there will be no instructing without their instructors and things of that nature.They may allow the instructor to teach within the club. However the flight school may not let him teach somewhere else. You are dabbling in dangerous territory though. Any time you do something sneaky or dishonest you are running the risk of losing privileges at both places. Both places have a business to run and they understand that you have a goal that you would like to meet fiscally and quality. There are also legal and insurance issues. Aviation is full of bureaucracy. It is easy to get in trouble if you're not careful. I would do everything on the up and up and make sure everyone is informed on what is going on. Aviation is such a small community. Keep all relationships positive, honest and professional because you will deal with them again one day.
The second scenario is more possible, however complicated and possibly expensive. Again, make sure everyone is informed on the situation and everything is clear, especially between the flight instructors. But using two instructors will most likely extend training and increase cost. The logbook would not be an issue as long as the instructor signing you off has documented where he personally instructed you are the required topics. This has happened before, and if everyone involved is understanding and accepting of the terms and conditions, it can work well. Don't try to hide anything or make sure you are covered
You will also learn how fellow pilots/friends view actions you are planning on taking. In other words if you are coming to my club to fly an airplane and you bring a guy to watch you fly. In doing so you are taking a job from my club CFI. It can be disespectful. Again, aviation is small an you will be found out. And if you get caught doing things of this nature, you will be held responsible to all party's you involved. Both Instructors, the flight school, the flight club, and any others that assisted you're plan. Because of the manor you treated the situation, when something goes wrong, you will not have those fbo's or flying clubs to support you because you have put them in an unwanted situation. Both business would be losing money and both instructors would create a a bad reputation and untimately lose business.
As for me, I understand what you want to do, and it can be tough to do. Just discuss the situations and get written approvals saying what you worked out is legit and everyone is in agreement. But really, juat keep it as simple as possibile. Avoid schemes and plans and I believe you will finish in shorter time and less money.
My best situiation is to find an independant GA aircraft that you could rent man to man and then pay the instructor. Just eliminate the flying club and find a plane for sale or rent, and se if you could rent, or block rent it.
**I hope this is comprehendable. I wrote it completely inebriated. Please be gentle in the mockery and hurtful shots at my manhood.