構【かま】う (= "care about", "mind", "worry about") can be used in the forms of both "～に構う" (intransitively) and "～を構う" (transitively).
For example, you can both say 「俺はお前に構っている暇がない」 and 「俺はお前を構っている暇がない」, and they're semantically the same! According to BCCWJ Corpus, "～に構う" is roughly three times more common than "～を構う".
You seem to know how to make causative forms from both intransitive and transitive verbs, but here's the summary. This question helps, too.
The causative version of "SがVする" is "SをVさせる" if V is an intransitive verb that does not take を. So:
琥珀さんと翡翠が 俺一人に かまっている (かまう here is intransitive)
→ 琥珀さんと翡翠を 俺一人に かまわせている (using causative form)
→ 俺一人に 琥珀さんと翡翠を かまわせている (swapping word order)
The causative version of "SがOをVする" is "SにOをVさせる" if V is a transitive verb. So:
琥珀さんと翡翠が 俺一人を かまっている (かまう here is transitive)
→ 琥珀さんと翡翠に 俺一人を かまわせている (using causative form)
→ 俺一人を 琥珀さんと翡翠に かまわせている (swapping word order)
In conclusion, the original sentence and your suggestion are both valid, and sound equally natural to me.