Today I got a call from a lady staff member at Strecker-Mayborn Museum, in Waco. She told me that Baylor University, the owner of Strecker, did have the saddle for a long time. She said that it had been donated it to the Layland Museum, in Cleburne, Texas, in 1996.
I then called the Layland. a lady there said that they do indeed have it, as well as his gun. She related that the saddle had been completely restored by an expert conservator.
I told her that some have questioned the existence of the saddle, and wondered if it could actually have belonged to Cleburne,
She said that it definitely was Cleburne's saddle, and that its provenance was well documented.
I am going to Cleburne to see for myself, and I will let you know what I find learn. This is going to be fun, no matter how it turns out.