Why e3? It doesn't help the Bishop at all, and the Viceroy is a weak piece. Moving a Knight out or preparing to open a line for a Rook might be stronger.
Pre-1500 opening strategy
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Regardless of the movement rules it's still all about mobility. Range of movement dictates attack, defense, and checkmate (even if in this game it goes 1 extra move and you capture the king).
So central space is still the most useful. Also, the least valuable pieces will still be the most useful in claiming this space as the powerful pieces are compelled to retreat when attacked by a lesser piece.
So moves like e3 and d3 followed soon by e4 or d4 would still be quite useful. Then find some setup which claims space while keeping the pieces in contact with each other and you'd have a basic and reasonable opening. Kind of like c4, d4, Nf3 Nc3 leaves no undefended pieces and claims central space I'm sure with minimal tinkering you could find a way to mimic that kind of coordination.
I suppose tactics would be much much less common... and no pawn promotion? I imagine many games would end in a draw. But anyway that's how the opening should still work.
I would recommend the reading of the first chapter of "The Joys of Chess" by Christian Hesse. I learned some excellent history of the early game of chess or shatranj as it was called back then. The openings seemed more similar to army strategy then the tactical game we play today, with most wins occuring by capturing all your opponents pieces rather than todays checkmate.
I've been thinking about the Xmas Eve game in 1026 between King Canute and Jarl Ulf that ended poorly for the latter. What would have been the openings for a game in which the pawn could only move one space, the viceroy (not the queen) could only move one space diagonally, and the bishops moved two spaces diagonally and were allowed to hop like knights over intervening pieces? In addition, there was no castling, plus the king would be actually captured. There was no pawn promotion or en passant captures, so the pawns wouldn't range that far from the rest of the pieces. Does anyone know of any strategy references for this version of game? Would the best opening move for white be e3?