(tx) Is There A Way To Court-order Paternity Testing As A Possible Biological Father Who Does Not Have Custody Of The Child In Question?

Edit: If the answer is 'yes', what is that type of filing / request called and with whom would we file it? The custodial / maternal parent lives in a different county than we do.

Location: TX, USA

The story:

There is a woman- we'll call her Josie- that my husband had a brief sexual relationship with at the beginning of 2019; a couple years before he and I met. Josie was also seeing other men during this time. Josie gave birth to a little girl- we'll call her Ana- and based on Ana's date of birth, it is not impossible that my husband- we'll call him Noodle- could be Ana's biological father. Noodle was made aware of the pregnancy shortly before Ana's birth and did provide some support to Josie after Ana was born in the form of money & diapers, but Josie started a 'serious' relationship with another man that she'd been seeing for we-don't-know-how-long before Ana was born and essentially told Noodle she didn't want or need his help anymore and to get lost.

About two years ago, Josie left a note in our mailbox claiming that Noodle is Ana's father and various "I wish you could see her" and "she's growing up so fast" sentiments. There was a string of text conversations that eventually lead to Josie asking for money and Noodle, being the sweet and sometimes foolish man that he is, giving her money multiple times under the condition that they do paternity testing that Noodle and I would pay for, which Josie agreed to.

We let Josie pick the AABB accredited laboratory that was most convenient for her, made the appointment when she said it was convenient for her, and showed up when we were supposed to. We have done this three times in the last year and a half and she has failed to appear each time. These are not the only agreed upon meetings she has fallen through on, and after the missed testing appointments, she goes radio silent for a couple months until she wants money again.

After her most recent failure to appear, Josie said she would go through the Texas Attorney General's Office to order paternity testing. We were perfectly fine with that, but now it's been nearly 6 months since then and nothing has happened.

The point:

We just want this to be settled. We are tired and anxious from dealing with a wishy-washy person randomly popping up in our life with the looming possibility of needing to pay child support in tow. If child support payments are what need to happen, that's fine, we just need to know. Noodle and I do not and will not be having any children together, nor do we have any intention of asking for custody or visitation rights, but Noodle and I both feel very strongly about financially supporting Ana if it is our responsibility to do so, and the combination of not knowing if Ana is Noodle's and whether or not we need to make major adjustments to our plans for the next decade is exhausting and has made us both so anxious.

Other information:

1) Josie has repeatedly made a big deal about not wanting to file for child support because she "cares about" Noodle.

I find it hard to believe that she cares about the feelings / well-being of a man that she had sex with a few times over 7 years ago more than making sure her children are financially supported if she really believed that Noodle is Ana's biological father.

2) Josie has 4 children in total: 2 were born before she and Noodle had a fling, Ana, and then her current youngest. She has custody of all 4, as far as I know.

3) Josie has been married since 2021, but is currently in the process of divorcing her husband.

This doesn't surprise me since he has a criminal record that includes multiple instances of intentional property damage, embezzlement, financial elder abuse, and is old enough that he has children the same age as Josie.

4) Josie has at no point mentioned she was / is married and / or getting divorced, or that she has a 4th little one. I found that out on my own through social media, and her husband's criminal record with a background check. I have not been able to determine how long the divorce process has been ongoing.

5) We have multiple written records of Josie claiming that Noodle is Ana's biological father.

6) I was able to confirm that Noodle is not listed on Ana's birth certificate as her father, but have not been able to fully confirm that no one else is listed as the father on Ana's birth certificate; Josie claims no one is listed as the father on Ana's birth certificate.

7) I have tried calling the Texas Attorney General's Office over a dozen times to ask if there is a way we can court-order the genetic testing even though we are not filing for child support since the child in question is not in our care. My calls gets dropped while on hold with a representative. Every. Single. Time.

8) Noodle has stopped sending Josie money.

9) When this all started, I supported Noodle in wanting to make sure Josie has what she needs to take care of Ana. Noodle and I were willing to give Josie the benefit of the doubt and give her some financial support until the paternity testing was done because he did have sex with her, and if it turned out that Noodle wasn't Ana's biological father, we spent a few hundred dollars to make a kid's life easier for a little bit. After two years of Josie being inconsistent, unreliable, and telling weird, easily disproveable lies to try and illicit emotional responses from my husband, I am so tired of her. I am tired of and stressed about not knowing if or how much of an impact this could have on my future plans with my husband. I am tired of and angry about someone stressing me and my husband out. I don't care what needs to be done, I just need to know what it is.

10) It would give me so much peace to be able to to file a no-contact order against Josie, but I don't think a judge or law enforcement would care, and mostly I worry how doing that could reflect on my husband / impact the determination of support obligation if Josie ever does get around to filing for child support and by doing so, we find out that Noodle is definitely Ana's biological father. (i.e. I don't want a judge to max out the child support obligation amount because a no-contact order made it look like my husband was trying to weasel out of his responsibility toward a child he helped create.)

11) If we had lawyer money, I wouldn't be asking y'all.