There are two issues - what does it mean, and why did Trollope deploy it here (sorry).
As a set phrase wrt women, it is condescending. It means the person knows the ways of the world, and isn't likely to be caught short. Moreover, they are quite willing and able to exploit those a bit less worldly, including those belonging to a higher social class. By itself, it is neither pejorative or approbative, but it is condescending.
As to why Trollope chose it, that's harder to answer. I suspect its use by Mr. Ingram is intended to say as much about him as about the pushy Miss Dawkins.
It establishes or reinforces a contrast and I think it gives the impression Mr Ingram, who has designs of his own, is perhaps over-eager to distance himself from Miss Dawkins in the eyes of his companions.
One reason for the condescending usage has to do with Waterloo. It seems all the old beggars in England claimed to have been in the Battle, and hence "Play the old soldier" and "come the old soldier" refer to deceit to avoid actually working or paying your own way. This is the sense that struck me most strongly.
COME THE OLD SOLDIER verb (also COME THE TIN SOLDIER / PUT THE OLD SOLDIER ON [18th century and still in use]: To deceive another for one’s own benefit, especially to avoid an unpleasant task [‘come the’ + Standard English ‘old soldier’; the skills of a veteran who, supposedly, knows every trick when it comes to avoiding onerous duties. Ware [[in Passing English of the Victorian Era] (1909)]] also cites the rash of beggars who proliferated in London after Waterloo (1815), all claiming to have taken part in the battle. Note nautical jargon soldier, a poor or lazy seaman, a shirker]
This is from Cassell's Dictionary of Slang, but I lifted it from Ken Greenwald's post here — Wordwizard
Thanks to @StoneyB's comment, I can add this —
It is interesting to note that ‘an old soldier’ had become a pejorative working-class expression for a scrounger
Rogues and Vagabonds: Vagrant Underworld in Britain 1815-1985
Since clicking through to these Google Books links can be problematic, you can bootstrap this by going to Google Books Advanced Search, and searching for "old soldier" in exact phrase and "rogues" in title.