Can I send the invitation/sponsorship letter today, knowing that my friend is going to apply for the visa after two months?
Yes (on the basis I know no requirement otherwise).
Would the invitation letter be still valid, specially the bank statements?
Is it going to increase the chances of rejection?
Yes, but can't be proven.
there will be a lot of extra/unplanned costs
So be it, explain what these are (presumably you will have convincing back-up for their exceptional nature). See What Is the Purpose of Bank Statements?. This may be a (rare) case for providing more than the minimum information required (that is, both the statements you would like to show and the more recent ones).
An 'open' invitation letter would not be as convincing as one that is specific. That is, Mahdi is required to attend meetings won't be as effective as say Mahdi is required to attend meetings in weeks 35 and 36 with our Technical Director, Sales Management and others in connection with our expansion plans for our Sudanese subsidiary. In the latter case, the invitation letter would cease to be valid after