If you are using the Beocenter to play LP's & 45's, and using your other turntable into the ' tape 2' socket on the Beocenter, you only need one set of 'speakers, no switching box required!
With regard to the extra turntable, the cartridge used will need to be changed to one with a stylus for 78's, as you say, a wider tip - and if you ever want to play 'vinyl' on this turntable, you'll need a second cartridge with a stylus for LP's. It will need a preamplifier to bring the output up to 'line' level; most give straightforward RIAA equalisation. As has been said - quite correctly - 78's were not RIAA corrected, but also early LP's had different correction curves from different record companies. In practice, most LP's are close enough to the RIAA curve to be acceptable, however, 78's are a very different matter. The ideal would be something like one of the Graham Slee preamps with switchable correction, but they are a serious investment... and in practice, many people in the 1950's when they were first playing both formats, simply used the tone controls on their amplifiers until they were happy with the sound. (Quad, for example, with their preamps, had a table of advised settings for the tone & slope controls.
I would start by using a basic RIAA preamp - which can be fairly cheap - and see how it sounds with 78's. Probably a bit bass-heavy & treble-light, possibly acceptable - if not you can consider another mortgage(!) to buy a switchable one. Because the 'Tape 2' connection is RCA ('phono') sockets, input from the preamp will use one pair of sockets, leaving the output pair for connection to your chosen recording medium, whether it be your computer or a hard disk or CD recorder.