After a bit of digging, I traced the exposition of your Theorem 3.14/15 first to Fedoryuk, then to Erdelyi. However Erdelyi's treatment of the problem is different than what you need:

The main purpose of the present paper is to supply explicit expressions for the error terms associated with the expansions of [4] from which realistic bounds are readily computable. The derivations of Erd61yi do not lend themselves readily to the construction of error bounds owing to the somewhat artificial nature of the neutralizer functions employed in the analysis. Our approach is 15ased instead on Hardy’s theory of generalized integrals [7], [8].

This is from a paper of Olver available here. While it deals only with the one-dimensional case, I would expect that to give you a decent starting point.

After a bit of digging, I traced the exposition of your Theorem 3.14/15 first to Fedoryuk, then to Erdelyi. However Erdelyi's treatment of the problem is different than what you need:

The main purpose of the present paper is to supply explicit expressions for the error terms associated with the expansions of [4] from which realistic bounds are readily computable. The derivations of Erdelyi do not lend themselves readily to the construction of error bounds owing to the somewhat artificial nature of the neutralizer functions employed in the analysis. Our approach is based instead on Hardy’s theory of generalized integrals [7], [8].

This is from a paper of Olver available here. While it deals only with the one-dimensional case, I would expect that to give you a decent starting point.