For this method to work, you really need to take m of the form 2^k-3^x with x>m/0.58 which we know is not possible. So it seems difficult to verify the diversity of residues.
A workaround is to choose m among the divisors of 2^k-3^x and such that m < 0.58x. For example, if we take k=249 and x=157, then m=67 is a suitable choice to check the method. Then we find that 11 residues are repeated twice among the first 66 residues. The first repetition occurs for d_{21} and we get all residues after 91 swaps. So it turns out that some repetitions cannot be ruled out. This is due to the fact that 67 is also a divisor of 2^{48}-3^{30}, 2^9-3^7 and 2^{12}-3^2. The last part of the proof is difficult to implement.
Maybe the value c=0.58 is simply too optimistic. Or maybe this method does not work at all and we should consider simultaneous swaps as in your third try…