"Rather than take a taxi, he chose to walk."
"Rather than taking a taxi, he chose to walk."
NOT A TEACHER
(1) I have found an ( not the) answer.
(2) Dr. John B. Opdycke in his 1941 and 1965 copyrighted Harper's English Grammar
does not recommend this sentence:
Rather than going with John, he decided to stay home.
(a) Dr. Opdycke suggests:
Rather than go with John, he decided to stay home.
(i) His reason? "to stay" is an infinitive. Therefore, it is preferable to match the other part of the sentence with an infinitive, too. (Grammar books call this parallelism.)
That is: Rather than (to) go with John, he decided to stay at home. (P.S. You must NOT use the "to" with the first infinitive; nevertheless, it is an infinitive. (Grammar books call it a bare infinitive.)
(3) If you accept Dr. Opdycke's reasoning, then it appears that he would suggest:
"Rather than (to) take a taxi, he chose to walk." (REMEMBER: Do NOT say/write the "to" with "take." It is only there in theory.)
(a) I checked several experts, and they would seem to approve of this, too:
Rather than taking a taxi, he walked.
(b) I suspect many native speakers would also use your second sentence of "Instead of taking a taxi, he chose to walk."
(c) Please go to the search box at this website and type in "rather than." You will find many helpful discussions.
Student or Learner