Besides consulting with a Texas lawyer to find out what the Texas law on such matters is (since a Florida lawyer may not know everything about Texas), there seems little you can do. And if, as you suggest, the amount in question is not worth the expense of legal endeavors then you may just have to let it go. I suppose the estate administrator could send a copy of the death certificate and the estate administration authorization / power of attorney, etc. requesting closure - but it sounds as if that may already have been done.
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You need to talk to your FL attorney and possibly a TX attorney to determine what needs to happen. It may be that the bank in TX will require you to raise an estate in FL which could be expensive. At minimum pay for a consultation, but make sure to obtain an accurate quote as to fees for any services and costs as the fees and costs could be a lot and it may not be worthwhile.
In Arkansas, there is such a thing as a "small estate" probate that costs very little to complete and is completed almost immediately after filing the petition. I do not know if Texas has similar probate proceedings. However, almost every lawyer in Texarkana is licensed in both AR and TX. I would contact one down there and find out if they can help you. It may not cost as much as you think if you have the option of a "small estate" probate action in TX like they do in AR.
This answer does not create a attorney client relationship and cannot be relied upon as a legal opinion as there are too many unknown facts to this matter to render an opinion at this time. Campbell & Grooms, PLLC http://www.campbellgrooms.com