As Jaap Scherphuis commented, if you change "terminal positions are losing positions" to "terminal positions are winning positions", then you are considering a game under the "misère play convention" (as opposed to the "normal play convention").
Now, if you have an impartial game (the sort of game you would apply Sprague-Grundy to) under misère play, and just want to see who has a winning strategy, then the difficulty of that depends a lot on the game you're considering.
If your question is just something like "can a tiny modification to the method of the Sprague-Grundy theorem handle all impartial games under misère play?", then the answer is "no", as in my answer to Nimbers for misère games.
More broadly, many papers have been written on handling sums of games that are coming from a single ruleset (e.g. Nim, or Kayles), both general theory and papers on specific games. Sometimes that's a straightforward analysis, and sometimes it's not. You can see a lot of these papers at miseregames.org.
The Sprague Grundy Theorem also tells you a lot more than just how to play a sum of Nim games and Kayles, it says that the nimber tells you enough information to determine who wins a sum of that game in combination with any other impartial games in normal play. However, there cannot be a similarly tidy result for misère games.
There is theory (see, for instance, section V.3 of Aaron N. Siegel's Combinatorial Game Theory) that lets you discard some of the information of an impartial game when misère-played in a sum of arbitrary impartial games. However, I do not believe this simplifies things significantly in any known nontrivial ruleset. For example, in my paper Equality Classes of Nim Positions under Misère Play, it is shown that: In the case of Nim, the only simplification this buys you is basically the ability to subtract 1 from any pair of odd-sized heaps (or add 1 to any pair of even-sized heaps).
