Some people claim you can reduce the bitterness of eggplant by slicing it then salting the slices, wrapping in a clean cloth, and putting some weight on it. The salt draws out moisture. It's possible a similar technique would work with your bitter melon, although I had never heard of it before your question. It might not save much time, but you wouldn't have to watch them while doing the salt treatment.
Assuming you're talking about ggaennip, also known as shiso, it's quite common to use it raw or salt-pickled. There are two common varieties, red and green-leafed, and to my taste there are some subtle differences between the Korean, Japanese, and maybe Vietnamese varieties.
In Korea and Vietnam, ggaennip is frequently used to wrap foods (ssam bap, for example, or the Vietnamese turmeric crepe). Shiso can be used to wrap foods, as a garnish, or chopped and served on some things, like grilled chicken (yakitori) as a flavor contrast.
I don't see much point in cooking shiso or yukari, but I've cooked the red pepper and salt pickled version sold in cans (or sometimes in the freshly made kimchi section) by adapting it into a sort of variation on the mediterranean dolma. It retains much of the flavor, but is different than the raw leaf and mellower than just the pickle it starts from.
I do appreciate shiso tempura, so I'm not dogmatic about not-cooking, and there is certainly precedent for some cooked applications. I've also made an agedashi-doufu, deep fried tofu in soup stock, which had shiso added before it was coated for deep frying.
Shiso/gaennip is now popularly used in a drink that has added lemon juice or citric acid, but I've only seen this as a concentrate or prepared beverage, so it's possible that it requires some industrial techniques to retain the flavor when making a syrup.
Dried salted shiso, called yukari in Japanese, is also used raw, but often mixed into warm rice either as a topping or to make onigiri.
Best Answer
According to a number of sources, the shoots are indeed edible, however all references I found were to "young" shoots, so I don't know about the mature vine itself.
The Curious Gardener says
Although the flowers, leaves and shoots are mostly used raw as salad greens, The Food and Agriculture Organization of The United Nations adds:
This North Carolina State University Article lists a number of cucurbits with edible shoots, including Chayote, Bitter Melon and a few types of Gourds.