What we know of the historical fencing sources from the early modern period is that opinions seem to differ! (If you know fencing instructors, this is the most unsurprising thing ever)

Liechtenaur (one of the earlier sources c. 1400) makes passing mention of teaching lefties to fence left-handed.

Thibault, writing c.1630, makes the surprisingly modern argument that left-handers were more difficult to fence because the other fencer is less used to fencing a left-hander than a right-hander.

But then you have de Viema, writing in 1639, going on an awesome rant about how everybody says lefties are more dangerous, but they're wrong, and how he fixes them into righthanders.

It has been the opinion among the vulgar that the lefty has an advantage over the righty in arms. The opinion has been well-received, because the vulgar teaching is (and has been) so far from reason that I find being a left-handed man better than being a diestro of the vulgar. Being a lefty is one of the greatest misfortunes that a man can have, and as I have proven in this science, the natural movements do not let us know this science. The lefty (a man we call backward) uses the movements of his defense with great profile of the body and straightness of the sword, and the right-handed man, by his own nature, places the sword obtuse, and the body squared, with which the lefty has an advantage. But the diestro that will come to know this science will easily be lord over the lefty, and thus I will give some blows so that the master can teach his disciple the mode that they have to have to play with lefties. These will be with the master taking the sword in his left hand, embodying the master, which is to provide the actions and postures that the lefty can make. Working it well, and giving to each blow the beginning and end that it deserves, he will see that the same blows will serve in favor of the lefty, only that they appear to be done opposite; these will be the generals that we deal with.

I am one of the men that have abhorred lefties most in this world. I am not scared that a man is left-handed, but in this I place the blame on his parents for not remedying it truly, as with this it will be enough in order to not be so.

We go such that if that a man was found left-handed, he must become right-handed, or at least be left- and right-handed. As much as it is a fault to be left-handed, it is an honor to be righthanded, and to do with two hands. If until now it has been difficult for a lefty to become a righty, in this book will be found such ease with little teaching and continuous use, that in two months he will have reason and way in order to be a righty.

In any case, since your online source didn't have a source for signing bonuses or lefties in sieges, I think it's likely just made up. It's difficult to refute as categorically never happened though, because the middle ages is a long time and we can expect different military men and fencing instructors to have different views in the same way as they did later in the Early Modern period.

Sources: Angelo, Sidney "The Martial Arts of Renaissance Europe"

de Viema's awesome rant is in Chapter IIII of Part Three online here in PDF, about p. 133