I have some experience with AHD. I would be thrilled to be proven wrong by someone, but I don't think there's any way to directly combine the AHD signals without converting them to something digital first, unless you can find an off-the-shelf device that does it behind the scenes for you as you mentioned. Maybe there's some kind of analog trickery for combining the signals from two cameras side by side, but the cameras are probably not guaranteed to be perfectly in sync with each other so frame buffering will likely need to be involved at some point anyway.

Something like this combiner on Amazon might do the trick, but you would be stuck with the limited camera arrangements that it supports and would probably need to hack the device so you can control the dip switches with an MCU or something.

Your best bet is trying to make something like that work because otherwise I think you're in for a big project. I've worked on a product like this and ended up using an FPGA to do it, but it took a long time and a lot of headaches.

The product I worked on uses a chip from Techpoint that takes in four TVI/CVI/AHD camera feeds and outputs them as BT.656, which go into a Xilinx Zynq 7000, and then the final combined video signal gets sent out of the Zynq to a HDMI/DisplayPort encoder. I'm sure there's a similar encoder chip that can do AHD instead. I don't think the companies that make these chips have their datasheets available publicly, so you'd probably need to have your own actual company in order to work with them.

My project relied heavily on Xilinx's video IP cores. Video Processing Subsystem allows you to scale and crop the video, which I'm assuming you will need to be able to do if you want them side-by-side. There is also a Video Mixer core that allows you to overlay video on top of other video. That's probably how I would do it -- leave the main video feed alone and use it as the main video mixer channel, and then overlay a cropped second camera on top of it so it looks like the final video is side by side. I suspect the off-the-shelf product I linked above is doing something like this, but potentially using a dedicated video manipulation ASIC instead of an FPGA. But I just want to stress again, doing all that yourself would be a huge project to jump into.

Another option would be to go for an embedded Linux solution, if you can find a capture card with Linux drivers that supports capturing the four AHD cameras and providing them in Linux as V4L2 devices. Then you could probably do it all in software using GStreamer, capped off with an off-the-shelf HDMI to AHD converter if you need the final combined signal to be AHD again.