When you wish to mention it a second time, you will use
Therefore, on first mention, you want to use what will lead you to that, which is your second choice:
cAMP response element (CRE)-containing gene
In the first choice, you have separated the abbreviation from its meaning with a hyphen and neglected to include a hyphen between the last element of the noun (CRE) and the participle (containing). In the third, you have included the abbreviation after a word that is not part of the abbreviation.
The only additional consideration is whether you should use a one-en dash (option-hyphen) rather than a hyphen following (CRE):
cAMP response element (CRE)–containing gene
Many style guides recommend the one-en when the first part of the compound contains multiple words that are not connected by hyphens, and I tend to do that when editing. However, the slight difference in length is often lost on readers and so that issue is not as crucial as just making sure the abbreviation follows the meaning immediately and that the hyphen or en-dash is properly placed. Again, that describes your second choice.
Here's a link to a page that refers to "Photos taken with a Global Positioning System (GPS)-enabled device":
and a link to another that does not capitalize "global positioning device," which, IMO, is the better way to go: https://www.highbeam.com/doc/1P3-1613660681.html.
There is no need to capitalize the first letters of a common noun that is followed by an acronym.