First, you have to keep in mind that for the first 6 years the Z was being made, not one of them had that piece you're taking out. It was added only for the 77-78 model years. So while it may not be a great idea to take it out, I really, really doubt it's going to have a big effect. If it were me, I'd just take it out and call it good.
Second, thicker does not necessarily mean better. For example, when installing a roll bar or roll cage, I believe the max thickness in the rulebooks for the plates is something like 1/8" - if not that, definitely under 1/4". And 3/8" is thicker than 1/4" lol. Did you mean 3/16"? But I can't think of a single place on the car (maybe the tie down hooks/plates in front?) where 1/4" steel is used.