Maybe buying the swappable switch Chinese boards with Blue switches, because you can easily remove and open the switches. Then lube them as klennkellon said. I think they can quiet a bit, but the said thing is, you won't get the stiffer spring from a Cherry MX White and the click mechanism is not the same. So, I do not think there is a cheaper alternative to the Whites. Matias Quiet Click is an option, but they are also not cheaper boards.
Aristotles are sold from ZealPC for $0.65, so if you buy them to swap an already bought keyboard, will be even more expensive then getting a keyboard with Cherry White.