As an interesting sidelight, earlier references do not use a sauce in making Welsh Rabbit, but, rather, refer to melted cheese.
For instance, in her 1745 The Art of Cookery Made Plain and Easy, Hannah Glasse presents this recipe:
To make a Welsh Rabbit
Toast the bread on both sides, then toast the cheese on one side, lay it on the toast, and with a hot iron brown the other side. You may rub it over with mustard.
Ms Glasse also has versions of Scotch and English Rabbit as well. Special for Ishbel, here are her instructions
To Make a Scotch Rabbit
Toast a piece of bread very nicely on both sides, butter it, cut a slice of cheese about as big as the bread, toast it on both sides and lay it on the bread.
Finally, she has one version that's similar to how we do it nowadays. However, according to Ms Glasse, it's a variation of English Rabbit:
Or Do It Thus
Toast the bread and soak it in the wine; set it before the fire, cut your cheese in very thin slices, rub butter over the bottom of a plate, lay the cheese on, pour in two or three spoonfuls of white wine, cover it with another plate, set it over a chafing-dish of hot coals for two or three minutes; then stir it till it is doen and well mixed: you may stir in a little mustard; when it is enough, lay it on the bread, just brown it with a hot shovel. Sever it away hot.
Interestingly, American cookbooks, through about the middle 19th century, do not use any of the rabbit terms. For instance, in the 1839 The Kentucky Housewife, Lettice Bryan presents a recipe simply called "To Toast Cheese," which is, with only a minor variation, Hannah Glasse's Welsh Rabbit. Mrs Bryan's recipe actually uses bread as a variation. Her main instructions are to toast the cheese on both sides, period. But she then points out you can toast slices of bread on both sides, lay slices of cheese on them, and melt the cheese with a salamander.
So what we actually have is a dish that started out more as what we think of as a grilled cheese (albiet, open faced) sandwich, then evolved into a gooey, saucy dish as it's known nowadays.