The proposed bill does not specify the designer. One can guess who the authors of the bill think is the designer, but no, we don't know who they mean by "designer."
Do we really have to guess? I think it is pretty obvious. The legislators proposing this bill are not Raelians. We are talking about the Bible Belt of the US of A.
Furthermore, even if the authors of the bill personally think the designer is a deity, this would not affect the constitutionality of the proposed bill.
Yeah, it does. If the purpose of the bill is to proselytize then it is unconstitutional.
The bill is not religious in nature, if we read it at face value.
Yes, it is. Intelligent Design is a religiously motivated political movement to get creationism taught in public schools.
Of course, this argument could be refuted, but there's nothing unconstitutional about a bill that uses weak arguments to support its purpose.
I think you need to review the Dover case:
Kitzmiller v. Dover Area School District - Wikipedia