Hello from Canada! Hoping to find some advice.

Both James and John are giving you good advice in taking small bites to get this job done. Cutting the wood in 1/4 " 'depths' and 'widths' will be easier on your machine. When you get to the point that you diagram in your drawing with the quarter-inch of material left to remove to get a squared off edge, you can – as James suggests -- flip the workpiece over and then use a bit with a bearing on the bottom of the bit away from the router base. That way your router base will ride on the bottom of the workpiece and the bearing away from the router base will ride on the edge that you have already worked, using it as a guide. You will need to use the other piece of material is a straight edge guide for this cut as the bearing on the bit will control the cut. It is possible to get a double bearing bit which has a guide bearing on both the top and the bottom of the bit. For example, see the Freud 50-501 Top & Bottom Bearing Flush Trim Bit which will cut 1 1/4 " between the bearings. A couple of other things that might be worth considering since I gather you're dealing with a couple of fairly expensive pieces of butcher block countertop. First, you may want to buy a scrap piece of 1 x 12, cut it in half and glue the two flat pieces together so you have a workpiece 1 1/2" thick and nearly 12 inches wide. Practice the technique of routing from both sides of the scrap to get a good feel for what you're doing. When you're doing that, and on your project workpieces, you might want to run two or three strips of blue painter's tape across the end of the piece you're using as a straight edge guide. That way you will have a workpiece that is slightly longer that you need, but one which if you remove the painter's tape for the "final" pass, you'll only be removing a few thousandths of an inch of material which will hopefully give you a very good cut and resulting edge.

Hello all! Thanks very much to everyone who's given your input so far. It's much appreciated!

I'll try my best to address every point.

The router I have is the Dewalt DCW600B. Here is a link that has the product info.
I have not purchased any bits yet but was planning on purchasing this Freud straight bit as well as this Whiteside flush-trim bit (bearings on top and bottom).

Unfortunately, I do not have access to a proper workshop or a table saw. The closest thing I have to that is a circular saw with the Kreg Accu-Cut guide as well as the Kreg Rip-Cut guide.
I thought about using those, but taking off just about 1/8" seems like it would be very difficult to do with the circular saw.

Making multiple small passes, plunging deeper each time was what I was originally planning on doing. I figured since it's 1.5" thick hardwood, the router will chug and I'll have to do many passes. I guess what I need to do is see exactly how square both pieces of butcher block are. My hope was that these were in fact made to be nice and square, in which case I can just use the one as a fence guide for the other.

I will try to take some photos once I finish work. We live in a small house so it will be a little difficult to navigate these pieces (they're currently leaning against walls). 😅

Will definitely be doing some tests with scrap pieces for sure!

Nibbling out in multiple passes is a good practice, especially with a compact router with a 1/4 inch collet. And there is not reason to assume the butcher block is really straight, so finding a really straight guide is a really good idea. The other problem that may occur is that most of the weight of the router will be unsupported so it will be amazingly easy to lit it tilt and cut a gouge in the edge.

So, two suggestions. you need a nice long straight edge to butt the router against, and that's long enough for your cut. I have a nice, solid, piece of aluminum extrusion, about 1.5 inches wide by about half an inch high in an L shape. Clamp it down and you have a reliable straight edge.

Second suggestion, pull the second piece parallel to your cut line so it supports the other side of the router. This will keep it from tipping. You only need an inch or so clearance and your router won't tilt.

Now, about the circ saw. How well that works for a project like this depends on having a really nice blade. https://www.amazon.com/Freud-D0760X-Diablo-Finish-4-Inch/dp/B001CZEU0S This is about $19 from Amazon.

To make it into an incredibly useful substitute for some table saw cuts, you need a long strip of 1/4 inch masonite (high density fiber board) about 10 inches wide. Attach the straight edge to the smooth side of the masonite double stick tape and some countersunk short bolts up through the wide side of the L extrusion. Leave plenty of room for the circ saw, which you will hold against the short rise of the extrusion. Now, put the saw against the "fence" and cut of the excess masonite. You now have a very straight cutting guide that will allow you to make many cuts as clean as a whistle.

Very useful for breaking down and making precise cuts in sheet goods. Just remember to keep the good side down! And if you're fussy, you can put a strip of green painter's tape on the cut line to prevent splintering. A more complicated way to avoid splinetering is to make a very shallow cut (1-2 mm deep is enough) and then do the final cut in one pass. Only one caveat, make sure the blade is exactly 90 to the base of the saw! Otherwise your joints won't fit or glue up. I think an 8 foot and a five foot guide will do a lot for you over time.

A table saw is a great thing to have, but this straight edge saw guide will cover you for a lot of projects. Don't forget to clamp the guide to your work piece.