Based on other questions you have posted, I will assume you refer to some jurisdiction in the U.S.
Is Joe allowed to refuse this offer and go to court anyway?
I wonder if this is legal since it may seem as though Joe is not
interested in the actual amount but simply is interested in the fight
and/or defaming the corporation.
Yes, it is legal, especially if Joe prevails in his claim of damages to his property. For instance, Michigan MCL 600.2591(3)(a)(i) awards attorney fees and other costs to the prevailing party if the primary purpose in the lawsuit is "to harass, embarass, or injure that prevailing party". But with Joe being the prevailing party, MCL 600.2591 is a moot point (that is, devoid of relevance).
Even if Joe did not prevail in the claim, it would be hard for Acme to prove that Joe's primary purpose was to harass or unlawfully injure Acme if Joe's position has some legal merit (see the statute at (3)(a)(ii)).
Acme Corp offers a settlement to Joe for $1500, on condition he signs
a non-disclosure agreement.
This portrays that Acme's settlement offer to Joe is not merely devised to make him whole. Instead, Acme is trying to obtain a very different benefit: Joe's silence. As such, Joe is entitled to refuse Acme's bid for non-disclosure, and commence proceedings in court. If Acme removed from its settlement offer the condition of Joe's non-disclosure, Acme would have more merit in arguing that Joe's lawsuit was frivolous.
Lastly, to clarify concepts, the veracity of Joe's claim implies that he is not defaming Acme. And even if Joe's claim were knowingly false, his judicial proceedings in and of themselves cannot lead to a viable defamation lawsuit because judicial proceedings are (unfortunately) protected by absolute privilege.