The SBC might take the G5 crown, but unless it either upgrades the bowl alliances it has or forces a change in conference opponents in those bowls, it will never change its perception. The AAC and MWC play P5 teams in bowl games while we as conference champs play the 3rd place team in the worse conference, CUSA. Just beating other G5 schools in bowl games and not even G5 conference champs or ranked G5 schools does nothing to improve perception of UL or the SBC.
Example - top teams in AAC and MWC, the 2 top G5 conferences. All wins by G5 schools (except Cincy)
Cincy v Alabama
UAB v. BYU (ranked)
UTSA v SDSU (ranked)
UCF v Florida
Houston v Auburn
Air Force v Louisville
Utah State v Oregon State
CUSA is arguably the worse G5 conference, yet got UAB to play BYU and UTSA to play SDSU, both ranked schools, and neither won their conference. Since both UAB and UTSA won their bowl games against ranked teams, perception is enhanced for both and for the AAC which they will be joining.
This falls on the SBC, not UL. The SBC is making great strides, but for the SBC to move up in stature, it needs to set up its top team(s) for a higher level of success to improve it perception and that of its top teams.