Yes training can be provided for aircraft under 12,500 LBS
Category 1 and 2 apply to training in aircraft over 12,500 LBS. Category one is the standard process for application.
(a) Category 1—Regular processing for flight training on aircraft more
than 12,500 pounds.
A flight school may not provide flight training in
the operation of any aircraft having a maximum certificated takeoff
weight of more than 12,500 pounds to a candidate, except for a
candidate who receives expedited processing under paragraph (b) of
this section, unless—
Category 2 does lay out a process for expedited applications that has some time requirements but section (iv) notes that you still must hear back from the TSA before training begins.
(b) Category 2—Expedited processing for
flight training on aircraft more than 12,500 pounds.
(1) A flight school may not provide flight training in the operation of any
aircraft having a maximum certificated takeoff weight of more than
12,500 pounds to a candidate who meets any of the criteria of
paragraph (b)(2) of this section unless—
(iv) TSA has informed the flight school that the candidate does not pose a threat to aviation or national security or more than 5 days have elapsed since TSA received all of the information specified in paragraph (a)(2) of this section.
(v) The flight school begins the candidate's flight training within 180 days of either event specified in paragraph (b)(1)(iv) of this section.
Category 3 applies only to aircraft of 12,500 LBS or Less and section 4 does indeed allow you to start training so long as it beings within 180 days of your paperwork being submitted.
(c) Category 3—Flight training on aircraft 12,500 pounds or less. A
flight school may not provide flight training in the operation of any
aircraft having a maximum certificated takeoff weight of 12,500 pounds
or less to a candidate unless—
(1) The flight school has first notified TSA that the candidate has
requested such flight training.
(2) The candidate has submitted to TSA, in a form and manner
acceptable to TSA:
(i) The information required under paragraph (a)(2) of this section;
and
(ii) Any other information required by TSA.
(3) The flight school has submitted to TSA, in a form and manner
acceptable to TSA, a photograph of the candidate taken when the
candidate arrives at the flight school for flight training.
(4) The flight school begins the candidate's flight training within
180 days of the date the candidate submitted the information required
under paragraph (a)(2) of this section to TSA.
So for aircraft over 12,500LBS you must hear back from the TSA in some capacity before training begins. While for aircraft under 12,500LBS you can start your training as soon as the paperwork is submitted.