The standard trick to quantify the joint continuity of a product operation such as $(A_1,\ldots,A_k) \to A_1 \ldots A_k$ (which is essentially what you are trying to do here) is to split a difference such as
$$ A_1 \ldots A_k - B_1 \ldots B_k $$
as the telescoping sum of $k$ expressions of the form
$$ A_1 \ldots A_{i-1} (A_i - B_i) B_{i+1} \ldots B_k$$
and then estimate each term of the latter separately by various standard inequalities, e.g.
$$ \|AB\|{op} |AB\|_{op} \leq \|A\|{op} |A\|_{op} \|B\|_{op}.$$
This will give some bound on the norm of the difference of the product in terms of the norms of the individual differences $A_i - B_i$.
A similar device will also control $A_1 \ldots A_{n-1} B_{n-1}^{-1} \ldots B_1^{-1} - I$.
