Should engagement photos be edited in one consistent style, or can I mix looks like black-and-white and moody edits?

Ultimately style in editing is an artistic choice. As such, there is no "correct" answer as to whether a particular type of shoot should be edited "this" way or "that" way. There are trends that come and go as to what basic style is popular for common scenarios such as engagement photographs. Personally, I tend to prefer more "classic" looks that will weather well as time marches on, particularly for "lifetime" events such as engagements, weddings, graduations, and the like. But some folks prefer the more trendy looks that others would consider too gimmicky. There's room for all of it in the great big world we live in.

As a business decision, though, things are a bit clearer. If you want folks to look at your work and say to themselves, "That's who I want to shoot my engagement session when the time comes..." then you're going to need to eventually pick a style and stay within shouting distance of that style with most or all of your edits, at least in the work you do for pay. You need to establish a look that is recognizable enough for folks to see your images and know you're the one who shot them.

Sure, you'll miss out on folks who might be more interested in another way of making the images look. But I think you'll find that if your edits are all over the place you'll miss out on almost everyone. This is because people will see a few images they like, but they will also see many more images in other styles that they don't care for as much. Will they be concerned that the shots they liked were just a lucky accident? Will they wonder what kind of images you'll deliver if they hire you? If they think they'll only get a handful of photos they really want from everything you sell them will that make them want to hire you? You need to give them confidence regarding what they can expect to see in the images you deliver if they hire you.

Building a brand means creating an expectation of what you will deliver. That means you need to narrow it down.

You don't need to do this immediately. In fact, you probably should take a bit of time and experiment with different ways of editing¹ until you can find a groove that you like, or one that energizes your friends and their circles of friends. But if and when you decide to get serious about marketing your services as a photographer you'll need to be able to present a portfolio that is more unified.

I think one thing that is perfectly acceptable is to finish a few images in monochrome, particularly those that have strong compositional elements that can stand on their own without depending on complementary and contrasting colors to guide the eye through the frame. But within the monochrome category, you again need to pick a general style and stick with it. Do you tone all of your Monochrome images in a heavy sepia cast? Or maybe a slight hint of blue toning? Or pure B&W with the RGB values the same for every pixel?

Your instinct regarding a "ring shot" that looks good in monochrome was spot on! Those more "traditional" shots that everyone expects to be a part of their package, the same ones their parents, grandparents, aunts, uncles, older siblings and cousins, etc. have are great candidates for the B&W treatment. You can work the shot from more than one angle and then do the frame that works best as a monochrome edit in monochrome while leaving yourself room to still do a color edit of one of the other shots that is different enough from the one you choose to give the B&W treatment that they aren't repetitious.

One thing I avoid is to deliver the same image in more than one edit (other than with regard to aspect ratios for different print sizes - there one has to be flexible). Sometimes it's hard to resist the urge to do a color and B&W of the same shot when both look really nice! But in the end doing that seems to me to show indecisiveness. This is also true of different editing style for color images. Trust your instincts and choose which way represents the brand you're interested in building.

¹ You need to not only experiment with an editing style, but you also need to experiment with different lighting philosophies. "Natural light portrait photographers" (that is, those who can't be bothered to learn about controlling lighting to give a consistent look regardless of the weather) are a dime a dozen. Photographers who learn lighting, develop their own style, and can shoot the same look consistently regardless of external conditions are the ones folks are willing to pay a premium to have shoot for them!