Do you mean something like the one linked below?

This is what we have. The specifications say that a 13.75 inch cabinet opening is required (our doors are a bit wider than yours, 16.25 in. but there's 2.5 in of clearance between the can and the other door.)

btw, I love this compost set up. The lid isn't very tight but I haven't noticed any smells over the hot summer. We tend to wait until the can is full before we dump it so some of the scraps sit for several days.

Here is a link that might be useful: rev a shelf pivot waste

Zelmar, that looks like a great option. Thank you!

So great there are others out there considering how to best store their compost bucket. I have been using a plastic pail under my sink with a lid for many years. The lid does not fit on tightly, sits well on the bucket. I take it out once a week or so (we have a lot of trimmings). Does get stinky - use Borax in the pail and takes care of the problem - and believe me, can be a real problem at times! But you should see the great compost we make.
Now, planning a new kitchen in over 20 years. I would like to incorporate an electronic composter that I read about. I think I read that it had been on Oprah's show. Have read reviews and sounds great. Anyone heard about it? Whatever I end up doing, the compost bucket has to be right next to or under the sink.... The bucket we have been using is so old that I cannot read how many gals it holds. I am guessing 5 gals. Those that they sell that hold 1 gal would never do for us, they would get filled in 1 or 2 days!

Here is the ad - NatureMill makes the new NatureMill Automatic composter that takes the pain out of composting. Composting takes kitchen scrapes and yard waste and turns them into super fertilizer, helping to divert waste from the trash pile. This product is ideal for apartments and non-gardeners.

Traditionally, you take your scraps to a compost box outside and mix them with grass, leaves, and other yard waste. You need to balance green items and brown items and wait several months for results.

This new appliance does it all for you. Just add food waste and it does the rest. No need to collect waste, hike out to the backyard, rotate stuff. In two weeks you have compost. It only uses 5 kwh of power a month, approximately $0.50 per month. This device is made recycled and recyclable materials and starts at $299.99 from amazon.

Anyone out there heard of this or considered it? Hubby thinks we should just go the way of old plastic bucket; I am so excited to try the new composter. Hope to get some reviews.

Our countertop people just put in a Blanco Solon. The kitchen isn't even done yet, and I can tell already that it is going to be one of the best parts of the whole kitchen! It sits beautifully, flush with the countertop, and is in the perfect place (next to the sink) where it will get a lot of use. They are pricey, but extremely well made. The whole thing is stainless steel, and has a bucket that pulls out of the unit.

Here is a link that might be useful: Blanco Solon

That Solon is wonderful! We have a stainless steel, glass- topped bin that is likely designed for flour or sugar (has a silicone ring on the rim). It looks fine, fills in a few days- so it never gets a chance to spill. Fine by me.

We have a Nature Mill and we are planning to install it under the sink when we renovate. It works well and makes lovely compost quickly and without odor or mess.

Just ordered the Nature Mill Pro Version. They offered it on the site down to $320 with same warranty. Will report back and let you know how we like. Have read tons of positive reviews (incl. doggonegarder's :) so decided to bite the bullet and spend the money. They offer an under the counter installation kit for around $45 dollars that would fit in kitchen_angst's parameters, I believe,
Will try the composter by itself then when new kitchen is in (rebuilding entire house, hopefully within a year) will have it installed under sink.
BTW, the Blanco looks AMAZING (curious, about how much?) problem for me is that it would take up precious counter space which don't have much to spare.... wrestling with how to incorporate a drainboard instead which takes up our counterspace, whether part of sink or have made into the granite somehow - need to see if there are posts out ther on this.... Good luck to all the composters our there!