Restrictions to counter machines capturing LBA
Programs with natural numbers as data and as instructions increment, decrement and zero test are know as register machines or counter automata. (Wikipedia cites several sources, usually I refer to Minsky). 

Their power is that of the Turing machine, that is, they accept recursively enumerable sets (strings or numbers).

Thus they are more general than the LBA, which is more a complexity class (as mentioned in the comments). The length of the space used for computations is linear, otherwise the LBA are Turing Machines.  

To translate that restriction to counter automata I would suggest to restrict the variables to at most exponential in the size (length) of the input. First, you can code such variable by markers on the tape. That would prove one inclusion. I expect that every LBA can be simulated by an exponentially bounded counter automaton, after all, the tape is just a number an writing on the tape is changing a digit. There is a standard construction that translates the two sides of the tape (right and left from the reading position) as two numbers, and moving the head is just a multiplication/division, deciding the element under the reading head is modulo counting. You need extra counters (variables) for these operations and in particular you need the zero test.