Navy dress swords have changed over the years but it seems likely from the evidence in Heinlein's letter that his sword was on the pattern of this 1872 sword. Swords of this style could still have been in use when his older brother was a mishipman. The 1872 sword does have a slight curve to the blade and a rapier point. Navy dress swords after this were very similar except that they have somewhat lighter blades that are completely straight.
However, despite Heinlein's written statement that Lady Vivamus is his dress sword the actuality is that the description in the novel does not match the Navy dress sword exactly. The primary discrepancy is his description of the guard as a "bell guard". The Navy sword clearly does not have a bell guard, not even a very shallow bell guard such as those found on Civil War swords. Also, the blade is light to "chop bone" though possibly it could if it were absolutely necessary, risking damage. Modern dress swords are not sharp at all, though one of this vintage might be.