Anything known about this variation of Diffie-Hellman key exchange protocol?
... instead of addition of random errors I'm proposing deterministic erasure of the lower part of the data.


Truncation instead of addition of noise is an optimization that some LWE schemes use already.


  ... I'm wondering if anything is known about the security of this protocol? Has anything been published about this sort of technique, or is there some obvious attack that breaks it?


The more pertinent difference in your proposal is that it uses a one dimensional vector. There does exist a dimension/modulus size tradeoff for the LWE problem. But the problem is very sensitive to the concrete parameters that you instantiate it with. 

An obvious algebraic attack might not be able to solve the problem, but you might be surprised at what the LLL algorithm can do. Understanding how to use LLL to actually do so is challenging. There are lots of papers about it.