.avi is no 'video format' just a (very old) media-container.
it contained older codecs, such as Cinepak, h263, Sorenson or the (in)famous DivX.
it does NOT support the actual h.264 (such as in BluRay or AVCHD).
a direct export into an .avi media-container is supported by MacOS only in Cinepak (stone old, no good quality) or .DV (SDef, PAL or NTSC)
so, what does your broadcaster want, when asking for 'avi'?
ask for codec, resolution and max. bit-rate.
• export from within FCPX as h264
• open that in free tool Mpeg Streamclip
• convert to h263 or Sorenson to get some quality, set resolution to requested format, save as mov
• open created mov with Streamclip again, go directly to 'save as' and choose from drop-down the avi wrapper
but again, you need specific settings from your broadcaster .....