Hoping that someone who was in a clearly controlling and abusive marriage and had fertility issues is miserable is honestly an awful thing to say, and i hope it's just because it's still raw and you're hurting.
Your husband is the one who made vows to you, not her, and if he has cheated on you with her then he is the one who has betrayed you. Her doing a shitty thing sleeping with a married man doesn't justify any abuse she's suffered like she no longer deserves sympathy.
At the heart of this is a child who needs and deserves supporting, financially at the very least. For whatever reason it's either suspected or known that her husband is not the father, and she is claiming that if he isn't, then it is your husband who is. To go via a solicitor (you can google the company to check their legitimacy, don't call a number or visit a website listed in the letter til you've checked via search engine and confirmed) will have cost a pretty penny, as well as doing it via official channels. This makes it seem much more plausible that there is a very high likelihood that your husband is genuinely the father, as opposed to it being a scam or her picking a random man she knows to accuse.
His behaviour isn't mitigating this either, refusing to engage, denying with anger etc. A truly innocent man would be horrified and desperate to prove to his wife that he is not a cheat with a secret affair child.
Whether there's a legal requirement for him to do the test or not, he should do one either to try and prove to you its a false accusation, or if it is a possibility, to take on his legal and moral responsibility for the child he has created. To prevent either side from using a dodgy/fake lab with false results in their favour, an impartial, accredited lab should be used for the test to get the real result.
In your shoes my marriage/family would be shattered. If he didn't do the test I'd never believe him that he hadn't cheated and possibly fathered a child. If he does the test and it's positive then he's cheated and that's unforgivable. Even if he tests and it is negative i'd drive myself mad over why she'd pointed the finger at him in the first place, no smoke without fire as they say, especially given how he's reacted.
However this works out i wish you and your children the best.