2 real options. North or South of Lake Vic.
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Via the North.
The big bus to Kampala Uganda passes through Rwamagana every morning. From Kampala get straight on an overnight bus to Nairobi. From there down to Dar where you can bus it again to the Songwe border post between Malawi and Tanzania (might have to change in Mbeya). This is the quickest route, but a bit boring.
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Via the South.
Buses leave all the time from Rwamagana to Rusumo border with Tanzania. Overnight in Benaco/Ngara and get the big bus to Mwanza where you get the train to Dar. You could forego the overnight in Benaco/Ngara and intercept the train south of Mwanza - problem being you have no idea when it's coming. From Dar you can bus it as above, or get the Tazara train to Mbeya and then to Songwe on a small bus. This is much more interesting. The trains are a good experience. Second sleeping most economical for comfort, especially if you are male (single sex carriages - ladies get the babies and the smelly food).
Bear in mind that the middle of TZ is scrub and largely (/seemingly) empty and has little transport beyond the train. But saying that, the overall route to Malawi you propose is well travelled and no bother. It just takes time...