Am I doing something wrong? I get 94.9% for 243/256

243 is pretty good. You have to go up to 3^17 = 129140163 < 2^27 = 134217728 to beat it:

     1                   3                   4  2                  1 75.0%*
     2                   9                  16  4                  7 56.2%
     3                  27                  32  5                  5 84.4%*
     4                  81                 128  7                 47 63.3%
     5                 243                 256  8                 13 94.9%*
     6                 729                1024 10                295 71.2%
     7                2187                4096 12               1909 53.4%
     8                6561                8192 13               1631 80.1%
     9               19683               32768 15              13085 60.1%
    10               59049               65536 16               6487 90.1%
    11              177147              262144 18              84997 67.6%
    12              531441             1048576 20             517135 50.7%
    13             1594323             2097152 21             502829 76.0%
    14             4782969             8388608 23            3605639 57.0%
    15            14348907            16777216 24            2428309 85.5%
    16            43046721            67108864 26           24062143 64.1%
    17           129140163           134217728 27            5077565 96.2%*
    18           387420489           536870912 29          149450423 72.2%
    19          1162261467          2147483648 31          985222181 54.1%
    20          3486784401          4294967296 32          808182895 81.2%
    21         10460353203         17179869184 34         6719515981 60.9%
    22         31381059609         34359738368 35         2978678759 91.3%
    23         94143178827        137438953472 37        43295774645 68.5%
    24        282429536481        549755813888 39       267326277407 51.4%
    25        847288609443       1099511627776 40       252223018333 77.1%
    26       2541865828329       4398046511104 42      1856180682775 57.8%
    27       7625597484987       8796093022208 43      1170495537221 86.7%
    28      22876792454961      35184372088832 45     12307579633871 65.0%
    29      68630377364883      70368744177664 46      1738366812781 97.5%*
    30     205891132094649     281474976710656 48     75583844616007 73.1%
    31     617673396283947    1125899906842624 50    508226510558677 54.9%
    32    1853020188851841    2251799813685248 51    398779624833407 82.3%
    33    5559060566555523    9007199254740992 53   3448138688185469 61.7%
    34   16677181699666569   18014398509481984 54   1337216809815415 92.6%
    35   50031545098999707   72057594037927936 56  22026048938928229 69.4%
    36  150094635296999121  288230376151711744 58 138135740854712623 52.1%
    37  450283905890997363  576460752303423488 59 126176846412426125 78.1%
    38 1350851717672992089 2305843009213693952 61 954991291540701863 58.6%
    39 4052555153018976267 4611686018427387904 62 559130865408411637 87.9%

Generated with this:

    max_use = 0
    for i in range(1, 40):
        trinary = 3**i
        power = math.ceil(math.log2(trinary))
        binary = 2**int(power)
        diff = binary - trinary
        use = (trinary / binary)
        max_star = ''
        if use > max_use:
            max_use = use
            max_star = '*'
        print(f'    {i:2} {trinary:19} {binary:19} {power:2} {diff:18} {use:.1%}{max_star}')