Isn't it too optimistic to think that selling Mane brings in a substantial amount of money that could be used to pay Kane? Mane won't be sold for nothing, but I doubt that there is any club currently considering to pay a significant amount to Bayern Munich as Mane also has an ongoing contract. If it is not Saudi Arabia, which club is willing to pay Mane a high salary? He is one of the best paid players and he could also choose to stay and accept the fight in the team while making a fortune even if he doesn't play. All these negotiations will be cumbersome and can take much longer.
Not just Sadio Mane but Bayern Munich can sell even more players to supply the money for hiring this player. I think since Tuchel joined Bayern Munich he didn't make some changes and now he can even sell some players he doesn't need for the next season this way there won't be any problems with hiring Harry Kane and paying the money to Tottenham.
However the Bayern Munich I know got enough money to even hire Kane without selling any player.
Bayern Munich has the problem that many of the players they have on their team can play a world class game and then play abysmal as if they have been out of the game for a year. Nagelsmann had that issue and tried to rotate , but then players like Sane or Gnabry or also Mane were all the same. Suck one game, be brilliant the next game. That is why it is not easy to decide whom to sell and whom to keep. I think none of them is steady enough to be playing for Bayern Munich.
Bayern is known for not making ill-considered moves in the market and I think they should stick to that strategy and not abandon it for the sake of shooting from the hip and getting Kane. But they might left without choices soon although it seems that Kane also wants to go to Bayern.