Fix mp3float Header missing without re-encode
Right now the only solution I found is a two steps process:
Extract the mp3 stream:
$ ffmpeg -i input.avi -c:a copy input.mp3

Combine it back into the original AVI container:
$ ffmpeg -i input.avi -i input.mp3 -c:v copy -c:a copy -map 0:v:0 -map 1:a:0 output.avi

Leads to:
% ffprobe -hide_banner -i output.avi
Input #0, avi, from 'output.avi':
  Metadata:
    encoder         : Lavf58.20.100
  Duration: 01:19:44.96, start: 0.000000, bitrate: 1211 kb/s
    Stream #0:0: Video: mpeg4 (DX50 / 0x30355844), yuv420p, 512x408 [SAR 1:1 DAR 64:51], 1071 kb/s, 25 fps, 25 tbr, 25 tbn, 30k tbc
    Stream #0:1: Audio: mp3 (U[0][0][0] / 0x0055), 44100 Hz, stereo, fltp, 128 kb/s