Hi,
NextStrain uses IQTree to do its tree building and ancestral state reconstruction. It is probably much easier just to install that and use it directly.
The advantage of using BEAST is that you can sample a distribution of plausible reconstructions rather than a single ML point estimate so it allows you to construct statistical inferences.
If you are interested in reconstructing synonymous and non-synonymous mutations on the tree you can use the codon models but there is also a fast approximate method that uses the nucleotide models but imposes a codon structure on them.
Paper is here:
In BEAUti, you turn on codon partitioning (in Sites tab, "partition into codon positions = 1,2,3") then in States tab, turn on "Reconstruct synonymous/non-synonymous change counts"). You can reconstruct the states at the nodes as well, the advantage that these will be maintained as sense codons. If you want to reconstruct everything then you can do "Reconstruct complete change history on tree" - but a warning this produces very large trees as every change is written into them as annotations. I recommend not logging these too often.
Andrew