I have come to the conclusion that it is unlikely Type XV had developed by the 13th-century, and are much more likely to be a 14th-century type.

First, the archaeological evidence for early Type XV is not particularly strong. To my knowledge, there are not many museums that have Type XV swords that have been confidently dated to the 13th-century. This is my understated way of saying I don't know of any. Even if there are some, they need to be evaluated in light of the argument made near the bottom of this post.

Oakeshott makes an argument for an early dating, but we have to be careful, because Oakeshott had a bias of trying to date particular sword forms as early as he felt possible/plausible. In the instance of Wallace A.462, Oakeshott might have thought the sword could have been 13th-century, but the cross is classic 14th, as attested by Type XV long swords. Additionally, A.462 looks like a lot of other known 15th-century swords, a fact that Oakeshott himself admitted, not to mention the maker's mark.

There doesn't seem to be particularly compelling evidence from surviving effigies for a 13th-century date, either. The myArmoury Spotlight article claims there is an identical sword shown (to Wallace A.462) on the effigy of Sir John de Hanbury, although photos of the effigy only show his scabbard, which does have the diamond shape. However, having seen scabbards with this shape reproduced by Tod for other types of lenticular blades, I do not think we can assert that the effigy proves the type exists. Other swords from Effigies and Brasses, particularly when we can see a photo, rather than a sketch, do not seem to confirm the type.

There are some examples of ambiguous swords from period art that are cited as being Type XV. This is possible, but uncertain. Some medieval manuscripts have beautifully detailed depictions of swords; others less so. In the absence of fairly credible archaeological evidence for Type XV, I would be reluctant to make a dating based upon art alone.

Beyond this, there is also an issue of logic that is not easy to explain. Knights during the 13th-century wore mail, and later on, mail and coats-of-plates. If Type XV were developed during the 13th-century and proved effective at defeating armour, one would expect Type XV to be widely adopted. However, this isn't the case. The overwhelming majority of 13th-century single handed swords seem to have been Type X.a, Type XII, and then later Type XIV. We also have some examples of swords like XI.a, and XIII.b. Further, it should be noted that while Oakeshott identifies XIV.1 in Records as being a Type XIV with a diamond point-section, most Type XIV do not have that kind of point-section, and therefore XIV.1 is almost certainly a 14th-century sword from around the time when Type XVI develop. That is to say, the overwhelming majority of 13th-century swords are swords with fullers and non-diamond cross-sections and point-sections. Had Type XV developed during the 13th-century, it would be very surprising for this type to be rare, since its ability to defeat mail and coats-of-plates would have been valued in wars and conflicts. Oakeshott's timeline demands us to expect that 13th-century Type XV were rare, outlier, experimental swords, not well-known or used, until they suddenly become much more popular during the second half of the 14th-century- despite being evidently effective, perhaps even more effective, than earlier blade forms for defeating mail and coats-of-plates. This does not make very much sense, especially when a key driver in the development of weapons was creating forms that could be more effective at defeating the common armours of the day.

Given these factors, it seems unlikely to me that Type XV is a 13th-century sword type. On the contrary, a more logical evolution is for Type XIV to have come first, then XVI, an experimental form blending a fuller with a diamond cross-section, followed by the mature Type XV after this.