score:3
Can I submit my brother's bank statement instead of mine?
Your brother is acting as the sponsor, so yes. But to be clear, both of you must submit bank statements. His are needed to establish financial viability on your behalf; yours are needed to explain your circumstances in Egypt.
will they compare?
They will refer to your earlier application and may compare. This should not be a big problem if everything is transparent. If you have large movements in your bank account, they should be accompanied by an explanation. They get very worried when people contemplate a trip to the UK and their bank statements show recent movements that are irregular. That sort of thing doesn't usually happen with genuine visitors and hence they start looking at V 4.2 (c)
is genuinely seeking entry for a purpose that is permitted by the visitor routes (these are listed in Appendices 3, 4 and 5);
What type of visa should I apply for?
You want to change from a visit of four months to a visit of a week. In this case it makes more sense to select the business visitor. Remember that the online form hasn't been changed yet so it doesn't sync up with the visas available under the new rules. So regardless of what you select, your application will be assessed as a 'Standard Visitor' governed by Appendix V of the rules.
and, bearing in mind the reasons given for the recent refusal, what documents I should submit?
The Visitor: supporting documents guide contains a thorough walk-through of what both you and your brother should submit for evidence.
That answers your question. If you are following through with the guidance and find yourself with an evidential shortfall arising from the guidance you can always ask another question.
Overall I think this plan is misconceived. Your GMC appointment is in 6 weeks. You have not yet studied the rules or the guidance and have not yet begun to assemble your evidence; it's further compounded by the logistics of having your brother send you his evidence from the UK so that his stuff will be fresh at application time. Surely it will take him some time to gather his documents, authenticate them and post them to you. Plus you need to get translations for your stuff. Plus you have a refusal to explain along with a compelling change of circumstances to compose and submit.
You may wish to ask your brother to instruct a UK solicitor to minimally provide you with a consultation and doc check.