I assume the cover is sagging due to the fact the water sits on it. If you can slope it so it runs off, that problem may be solved.
Not sure how much slope you have now, but if you dont have much slope now, you may boit be able to add enough wood to the one side on the top to make it work and look decent.
can you lower the beams on one side to lower one end of the pergola to get the slope?
Not sure where you are from, but up here in Canada I was told as soon as you add a roof to a pergola it is considered like a covered deck and you need pilings under it...not sure if you have that or need it, but might want to check into it......taking roof off and getting an umbrella maybe the cheapest route if you dont have pilings but ended up needing them