The halting problem of Turing machines in view of enumeration of initial tape configurations
The Halting problem is stated in terms of any admissible numbering. There are effective (i.e. Turing-computable) compilers between all such numberings, so the Halting problems are all the same. That is, in particular not decidable.

There are numberings of all partially-recursive functions for which the Halting problem is decidable -- of course. For instance, give all "yes"-instances even and all "no"-instances odd numbers. Such a numbering would certainly not be admissible; in this case because it would not be partially recursive.

Thus, it comes down for you to show that your numbering is admissible or not. If not, the Halting problem may be decidable but the numbering would be ... not very useful.