The FARs do not say that the takeoffs and landings must be done in an N-registered aircraft. It does not say that you have to be PIC for those takeoffs or landings. It just says that you must be "sole manipulator of the controls" during those takeoffs and landings.
Go find an aircraft, get another pilot or instructor to go with you, but you perform the landings and takeoffs. As you can see, it says you can't have people on board except for those "necessary for the conduct of flight". So... another pilot who is current and legal to fly the plane, an instructor, etc...
FAR 61.57
Recent flight experience: Pilot in command.
(a) General experience.
(1) Except as provided in paragraph (e) of this section, no person may act as a pilot in command of an aircraft carrying passengers or of an aircraft certificated for more than one pilot flight crewmember unless that person has made at least three takeoffs and three landings within the preceding 90 days, and--
(2) For the purpose of meeting the requirements of paragraph (a)(1) of this section, a person may act as a pilot in command of an aircraft under day VFR or day IFR, provided no persons or property are carried on board the aircraft, other than those necessary for the conduct of the flight.