How do I approach a geo-location algorithm with two lidars?

I want to create a device to map caves in 3D in a cheap and fast way. For this I am planning to use two 2D lidar sensors and a digital compass. Since there is no GPS available in caves, I need an algorithm to predict/calculate the current geo-location of the device in the map that I am creating by walking around and measuring.

This graphic represents the two lidars (red) mounted on the device, one vertical and one horizontal:

I was thinking abot using a small LSTM model for each lidar to predict the movement up/down/left/right and forward/backward/left/right every few lidar scans (the freq. is 14 Hz).

The input data would be the last 10 lidar scans [10, 90] (scans, distances) for each lidar and the output the position change in cm in the different directions [4] (forward, backward, left, right) for the horizontal lidar and [4] (up, down, left, right) for the vertical lidar.

Afterwards I was planning on combining the results with the compass.

I don't know if this is a good approach or if there are any better algorithms oder architectures for this problem.

What would you suggest?