Corporate pilot reported while being vectored to the final approach course; ATC was late in giving a turn to join the localizer resulting in the aircraft flying through the final approach course and having to rejoin the localizer from the other side.

Narrative

Event Summary: On Day 0; IFR flight arriving DFW in Aircraft X at approximately XA:51 hrs; we were receiving radar vectors from TRACON (Lone Star Approach) for final approach to land; advised to expect 35C. Our last heading given was 300*. After multiple; continuous exchanges with other aircraft; ATC instructed us to intercept the 35C localizer and track it inbound. Due to the close proximity to the localizer course when the last instruction was given; it was not possible; without exceeding the normal operational capabilities of the aircraft to immediately comply with said instruction. We thus momentarily passed through the 35C localizer course and continued our turn to re-establish ourselves on the 35C localizer course. TRACON queried if we had flown through the localizer course. We replied we had and that we were unable to turn that abruptly in our aircraft. After we reestablished ourselves on the 35C localizer; we were handed off to DFW tower for a continuation to land.Chain of Events / Human Factors…ATC controllers at DFW / TRACON / CENTER do a great job of handling an enormous amount of traffic. We do not have intimate knowledge of scheduling or staffing issues during the current government shut down; but our controller; overworked perhaps; handling more aircraft than normal may have not realized our position / close proximity to the localizer course when issuing his last instruction for us to intercept said course.