That sounds like a really hurtful situation. Normally a court Order for a DNA test is begun at the Domestic Relations Office (DRO) which I understand from your summary has already refused to do one because of the child's age, that support has been paid for a long time and that an acknowledgement of paternity was signed at birth. The likelihood of having a court ordered DNA test seems small under these circumstances. but if this is really disturbing him he could consult with an experienced family law attorney to review his particular situation, determine his options. and .then weigh the pros and cons of proceeding down that path.
This answer is only for informational purposes, is general in nature and does not constitute legal advice. This Q&A forum does not in any way create an attorney/client relationship. If you have a legal concern that is important to you, I urge you to consult with an attorney as a failure to take prompt and appropriate action on any rights that you may have could be prejudicial to them.
It's called the Presumptive Father Doctrine. Even if the child is not actually the husband's, if it was born during the marriage, then it is presumed to be the husband's child. Also, it sounds like the passage of time has precluded any further look into paternity at this point. You could speak with a reputable attorney about looking further into this, but it would all depend on the specific facts of the case and how much time has passed since this became an issue.
Attorney answers to questions are for general purposes only and do not establish an attorney-client relationship. This answer is meant for informational purposes only and should not be construed as legal advice within the bounds of a professional relationship. It is always best to seek counsel with a competent attorney experienced in your area of issue and fully informed about the facts of your case.