You can use an LLC, Corp, or a Trust to be the owner of NFA items, including machineguns. This does mean that there will be no CLEO sign-off requirement. Most folks are now using Trusts - cost a little more on the front end to have them done right, but there are no year after year continuing registration or annual report fees and they are private.
However, if your CLEO will sign off (many will, I understand, in Texas), I would check into that first.
You should know that using a Corp, Trust or LLC is not really "skirting the CLEO sign-off". In states where NFA items are legal to own, the CLEO should sign Form 4's, but there is no penalty if they refuse. So, many simply refuse. CLEO's are skirting their responsibility to sign-off on Form 4's. The reason this requirement is there is not for them to act as gate-keepers of who can have an NFA item (The ATF and FBI handle this), it's so that the chief law enforcement officer of each county is aware of who owns silencers, machine guns, etc. in their jurisdiction. When they refuse to sign-off on a Form 4, they are doing the old Sargent Schultz "I know nothing, I see nothing" routine.
Also, NFA laws, specifically the '86 machinegun ban, are some of the most poorly written laws out there. And ATF's "Rules" appear to be written by "special" 3rd graders on crack, i.e. are often over-reaching and very inconsistent to the point of pointlessness and absurdity.