I can't give you an answer that is well-adapted to Lawson and Michelsohn's formulation of the pseudodifferential calculus. But here's how this sort of argument is supposed to go: two elliptic pseudodifferential operators with the same principal symbol differ by a smoothing operator, and smoothing operators are compact (one often proves this by showing they are Hilbert-Schmidt). By Atkinson's theorem, two Fredholm operators which differ by a compact operator have the same Fredholm index.
So the real point is to understand why the symbol of an operator characterizes it up to smoothing operators. The problem is that you can have a lot of flexibility in exactly what class of operators you decide to allow to be called "pseudodifferential", and so your proof of this fact has to be well-adapted to your specific construction. But I don't think that there is much controversy over the statement that whatever algebra of operators you consider to be the algebra of pseudodifferential operators there should be a symbol map whose kernel consists only of smoothing operators (or, at worst, compact operators). I unfortunately can't help you verify that Lawson and Michelsohn's choice meets this criterion.