It's pretty unclear what you're describing, providing an example of a lake with this feature might help.

Here is what I get from this: Picture a nice round lake at the surface shaped like a vertical hourglass in depth, thats is the "gets wider" part and what i think OP is asking about.

it is an interesting question, and is a quite possible shape. The first thing that comes to mind is the lake in Louisiana that flooded a salt dome underneath it, generating not an hourglass shape but a huge flooded cavern under the lake. This is at least a similar layout to the question.

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I don’t know any specific term. A lake like you describe is a rare thing. They might occur in karst terrain, but the size of any such lakes are fairly small. The largest cenotes (flooded sinkholes) are a few hundred metres across and won’t necessarily have a below the waterline undercut.

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Like this (the top is the surface):

 \             /
   \         /
     \     /
     /     \ 
   /_______\

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Right, I understand what an hourglass looks like, the point is that this type of question probably stemmed from a particular example and so providing that example would help narrow down specifically what OP is asking about.

The only hourglass shaped lake I know of is Warm Mineral Springs in FL. It has that shape due to a sinkhole when ocean levels were lower. With a hotspring erosion setting off the process. https://en.m.wikipedia.org/wiki/Warm_Mineral_Springs_(spring)#:~:text=Warm%20Mineral%20Springs%20is%20a%20sinkhole%20formed%20in%20carbonate%20rock,metres%20(236%20ft)%20across.

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You are correct, there is a specific example. I won't offer up the name for online safety reasons. I'm not affiliated with any organization and this is a passion project.

The lake in question, like the one mentioned in Florida, is mostly spring fed. The rock type is much more dense. It is classified as oligotrophic despite its boundaries having been deforested, used for agriculture for over a century, and summer home development. I find that bizarre and am attempting to gain a broader scope of understanding of the systems before possibly spending next winter counting diatoms from core samples.

Furthermore, I suspect the vast majority of bathymetric map data was collected via measuring a rope with a weight at it's end. Such a methodology would overlook the characteristic I've been referring to.

Lastly, I hypothesize lake bottoms which share this characteristic maintain other anomalies to limnology models.

The only thing I can imagine is karst terrain (limestone erosion) causing a sinkhole to form. Typical lake forming processes shouldn't result in this.

Grain of salt; I only have an undergrad in geology, and hydro was not my jam.

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there is myth of similar underwater caves where the "lake monster" lives

Bathymetric mapping techniques using either sonar or the interesting "rope with a weight on the end" method you mentioned do not capture overhanging features well. You would need to collect 3-D data like with an AUV or ROV to produce a model, and a standard bathymetric projection map would not be able to display it in any case.

I have been in a submersible and encountered large overhangs not shown on maps even in areas that have extensive data. It's unnerving.