Terve 
Yes, an MMC4 (or any MMC from 1 to 5) would fit your Beogram 6500, and it is quite likely that the poor sound quality is because the suspension in your old cartridge has failed. If you look closely at the stylus at rest, does it exit the body of the cartridge straight and centered in the opening?
The original idea of the MMC cartridges was that they are integrated with the stylus, and are always replaced as a unit. However, Axel Schürholz (I'd rather prefer - perhaps himself as well - that people referred to his services with the company name, schallplattennadeln.de, however awkward it might to spell 
) is able to replace the suspension and stylus with a new one.
I noted that cartridge you're referring to myself as well. I think it is a little pricey, especially since the seller doesn't say which stylus was installed, as the prices vary. You probably have an MMC2 in your Beogram now? You could just as well send it to be repaired to the same place and probably have a better end result. Prices start at 89,- for the simplest, spherical needle. I chose an elliptical for mine and I'm quite happy with it. At least never throw away your old cartridge!