You have done an excellent job of searching probate and tax records for Texas. Just a question or two.
1) If Juliet is indeed your 'Julia', she was born about 1846 +/- 3yrs. Do you know where your Julia was born? What about her parents?
2) I gather there were no names on the will of Robert F. Whitaker I. Were there any other Whitakers in Bastrop County?
a) He inherited his 'slaves', perhaps from a parent, grandparent, sibling, or in-law (for his wife)
b) This transaction could have taken place 'out of the state', or in another county, before moving to Bastrop or Red River counties.
c) Since you believe he acquired his 'slaves' between 1836-1840, he could NOT have acquired Julia during this period. You need to move BACK one generation. [Could Julia have been born to someone he DID acquire during those 4 years?]
d) Was Julia born in Texas? Where were her parents born, especially her mother? Any Whitaker 'kin' around?
e) Where was Anna born? What about her parents?
f) Where did Robert F. and Anna meet or marry?
g) Where were the children of Robert F. born? Where did they marry, especially the daughters? This may indicate a possible migration pattern.
h) Is it possible that Robert F. made a trip to Texas to buy land, then went back home to move the family and household down to Texas?
i) Did Robert F. spend any significant time on the 'trail' - in one or more states before settling in Bastrop or Red River county? Were their any relatives migrating with him or living along the trail?
j) Is Julia listed in any other records, perhaps with a parent or sibling?
If there are no other Texas transactions involving this Whitaker family, you need to expand the search. At the very least, you may need to reach further back in the tree. Were the ancestors of Robert F. Whitaker I slaveholders? What about those of Anna (was she Ann Weir/Ware)?
I know you probably didn't give ALL your information, dead ends, or lessons learned. Perhaps you have already considered one or more of these possibilities.
Sometimes it helps to review what you have, study the TIMELINE, and fill in gaps using resources and connections you may have missed.
The library has real life Case Studies (see link). Search through the 'Articles and Papers' section. The Beall Case Study is particularly interesting. The Courthouse Resources paper is a refresher course on available records.
There are some other good resources on the Slave Forum.
Read through - and post there (or re-post) if you would like additional suggestions.