Issues after replacing the BCM?

Hello, yes that is a burn in that circuit, this can happen if there is water intrusion into the BCM's case. Where does the BCM live in your vehicle? The real issue is that a component may have shorted out causing this burned trace on the circuit board.
Also, I don't see any 4-cylinder 2006 Colorado's. Can I get your VIN number to look the vehicle up?
What I do in these situations is use a multimeter and find out which pin on the connector that trace goes to, that way I can identify which circuit it is and what components are on that circuit. You'll want to check the components before replacing the BCM because you don't want a short to fry the new BCM.
And I did find your vehicle, sorry about that.
Very wise of you to take a look at the circuit board. It looks like there are possibly 2 or 3 traces that have burned on the board. Looking at it closely. That is an open circuit right now, if you can follow it and see if you can find where it goes, I'm looking at the wiring diagrams and the BCM communication wires are involved with the PCM in turning on the A/C compressor. Will post the diagrams.
The first two diagrams are the power feeds for the BCM, diagram 1 has the fuses and diagram 2 the ignition switch feeds. And a checksum error means they couldn't communicate with it. There was no saved data they could read, probably because they couldn't power it up possibly. I try to repair these types of components in my spare time. But that circuit shorted out either because of moisture getting into the casing or a component shorted out externally and took out the BCM.

Well, are you looking to get it fixed, obviously the circuit board has a burned trace or two on it, And it looks from a close up picture, to be a short circuit, most of the time when a trace on a circuit board gets corroded from water intrusion or a capacitor leaking its more of a greenish color. That looks like it is burned the circuit. So, for example, on the last engine computer (ECM) I had that failed, it was a shorted-out ignition coil that allowed direct voltage and too much current flow into the ECM and blew apart a mosfet (the driver for the ignition coil). Since your radio and your A/C compressor are not working and they state it was a checksum or data issue, and both the radio and A/C compressor do communicate on a 2 wire serial data network with the BCM being the Gateway module between the modules that control both of those. I'd say that's why they are not functioning. Below is a wiring diagram of the serial bus network and you can see on the right, the BCM is the gateway or link between most modules on that network. So, first thing is to check the fuses for the BCM and see if any of them are blown in the first diagram I posted above. Is this the new BCM that has got the burn mark on it? You were told it needed to be replaced, it got replaced and then the new one burned out too?

The last diagram I posted just shows that all the purple wiring runs through the BCM and then out to other modules. So, with the BCM not being in its place there is no communication getting through. I just need you to check the 3 fuses at the top of this diagram, and no, if I can re-solder the trace and fix the circuit on the board it shouldn't need any reprogramming, it should have retained all its original information. The soldering isn't a guarantee fix either, and thanks for the extra pictures, I'm going over them now, I don't see any other damage, but we wouldn't' know until the burn it repaired and I see if I can communicate with it. My only worry is that larger chip in the 1st picture on the bottom left. It appears that is where that circuit runs too. It's most like what controls whatever component is on that circuit. But if you notice any other burn marks let me know. There is a chip shortage right now that has been going on quite a while and a lot of people are waiting for new engine computers and modules, I haven't looked at the price of one of these BCMs, but I'm not sure how that company missed the huge burn on the side of it. My guess is they never bothered to open it up, they just plugged into it and got the checksum error. Hopefully they didn't charge you anything for that.
I did find that part number for $400.00 for a new one. A lot of other places seem to be out of stock so far. This isn't something we normally do here on this site, we're more of helping people diagnose their own vehicles, so I would have to okay it with the Boss man. But again. I can't guarantee anything.

I saw module experts, they cost a lot too. And yeah, a mobile tech would have to come out and program it. I'm not sure what that would cost. The fuses are TBC Fuse 34 (10amp), Turn/Haz RR fuse 28 (15amp), and Turn/Haz FR fuse 37 (15amp). $230.00 for doing nothing, that's not right.

By this picture can you tell me which connector is C1, C1 is the connector that clips sideways. So, the spring-loaded white hold down would be at the bottom, is that correct? I'm just trying to determine which connector the burnt trace goes to. I think the angled section faces down?

This is correct though, that is the top. I've been going over these pictures some more, it does look like there are a couple or maybe even 4 that ended up getting burned and the traces run down that angled piece and looks like they are from the C2 connector. The C2 also has the Ambient light sensor Ground, Ambient light sensor signal, Accessory Voltage. C1 has the A/C Request signal pin, and although I don't see any pins labelled Radio, the Radio is basically its own module, so it might need some type of wake-up signal from the BCM that it's not receiving and that's why it's not turning on. Everything works on networks now, just like a home internet system. Even if you end up deciding to go with a used BCM, I'd like to give it a try at fixing those circuits. If I can I'll send it back to you and you'll have a backup BCM. I hope some shorted component did cause this though. I always try to figure out what caused a failure like this. Especially a burn like that. I wouldn't charge you anything. You paid enough with g7.

I sent a message to the boss, just to make sure it's okay with him first. If you do grab a used one. I'm sure there are mobile techs in your area that can come program a BCM really quick. Are you out in Hawaii?

Hi there, alright I got the okay to try to fix it. If you want to give me your email, I'll contact you privately. I'll give you an address to send it to.

Okay, email sent to you.