Bought a Kohler Drop-In Stainless Steel Kitchen Sink. Plumber and vendor indicated no issues after providing dimensions of the cutout.

A few months have passed and ready to install. However, the clips that come with the sink and attach to the underside channel can only secure the back and sides. The clips cannot clamp down the front of the sink since there is simply no space in the counter to accommodate based on the design. Also, there is no possibility of increasing the cutout size to accommodate the front clips.

With the sink in place without anchoring with the clips, the four corners are on the counter and the center of each edge is a bit raised. This is most pronounced at the front of the sink.

Given the above, is it ok to secure the 3 sides and just use silicone adhesive like Liguid Nails All Purpose Adhesive Caulk (http://www.liquidnails.com/en-US/products/caulk-AC138) around the lip of the sink, but just putting a lot extra at the front to fill the space?

as long as it is mildew resistant it will be a good choice.
lay down a good thick bead all around the opening and smush the sink rim right into it. then tighten the clips you can reach.

clean-up with a damp cloth before it sets..

What type material is your countertop?

Agree, a good quality mildew resistant clear caulk will help "hide" most of the gap. but with SS sinks, they can be bent slightly if you have the patience and a "blacksmith's eye". Drop the sink in dry, no clips, and use little pieces of painter's tape to mark where the corners touch the counter, then cut some scrap 2 X 4 pieces of wood about as long as each gap is. Lift up the sink and slip the wood strips under the gaps. Might need a helper to hold the sink down, and wear some heavy duty work gloves to push down on the corner spots with the tape on them. (easy does it, a little at a time). This will help get a flush fit all around on the dry fit, then use caulk and clips to keep the sink from shifting until the caulk sets. No need to over-tighten clips.