One of the meanings of disappoint is to belie expectation. Since not all expectation is hopeful, and since the lawyer had indeed expected worse "from gentlemen in [his] position" and had learned not to depend on them, not to "found" his expectations on them, the upshot in this instance is an agreeable form of disappointment. He is happy to say he had expected worse but was wrong.
In the 19th century, especially in legal contexts, the verb disappoint could mean to undo something, such as a will, or a plan, and developed the sense to undo or thwart a normal or expected course, or to run counter to expectation, and it could be used without the sense of emotional dejection that accompanies the use of the word nowadays.
Stevenson's lawyer is using the word as a lawyer might have used it in the 19th century
Nowadays, disappointment is used 99.44% of the time in the sense of having one's hopes unfulfilled.
Here's an attestion from The Autobiography of Adin Ballou (Lowell, Mass, 1896).
... I am also happily disappointed in the place. Those who have spoken against this city have misrepresented it. It is superior to Boston.
Here are a couple of attestations from 1875, testimony given before the GENERAL ASSEMBLY OF ILLINOIS, AT ITS TWENTY-NINTH SESSION.
In what respect are you disappointed in his or her progress since he or she came to the institution?
In no respect are we in the least disappointed but on the contrary found the improvement beyond our expectation for which we are very grateful to you and your noble institution... In his personal appearance and his regard for cleanliness and his idea of things and places also in his dressing and undressing of bimself in which he has improved more than we expected, so that we are gladly disappointed in the progress he has made ...
... I am indeed very happily disappointed for she has made greater progress than I ever thought she would be capable of making ...
And here's one from the introduction to The Complete Poetical Works of Robert Burns... with A Life of the Author, James Currie, ed. (Philadelphia, 1844)
The sense of his poverty and of the approaching dis tress of his infant family pressed heavily on Burns as he lay on the bed of death. Yet he alluded to his indigence at times with something approaching to his wonted gaiety. What business said he to Dr Maxwell, who attended him with the utmost zeal, has a physician to waste his time on me? I am a poor pigeon not worth plucking. Alas I have not feather enough upon me to carry me to my grave And when his reason was lost in delirium his ideas ran in the same melancholy train. The horrors of a jail were continually present to his troubled imagination and produced the most affecting exclamations.
On the death of Burns the inhabitants of Dumfries and its neighbourhood opened a subscription for the support of his wife and family. The subscription was extended to other parts of Scotland and of England also particularly London and Liverpool. By this means a sum was raised amounting to seven hundred pounds and thus the widow and children were rescued from immediate distress and the most melancholy of the forebodings of Burns happily disappointed.
