Unless I have missed another relevant clause, it seems to me that one could train in an ATO on your own Tiger Moth (Annex II) and get your EU licence providing you take the skill test on the Tiger Moth or the same CLASS of aeroplane. Thereafter, if you want to fly something different you need to comply with FCL.135.A
ORA.ATO.135 Training aircraft and FSTDs
(a) The ATO shall use an adequate fleet of training aircraft or FSTDs appropriate to the courses of training provided.
AMC1 ORA.ATO.135 Training aircraft and FSTDs
ALL ATOs, EXCEPT THOSE PROVIDING FLIGHT TEST TRAINING
(a) The number of training aircraft may be affected by the availability of FSTDs.
(b) Each training aircraft should be:
(1) equipped as required in the training specifications concerning the course in which
it is used;
(2) except in the case of balloons or single-seat aircraft, fitted with primary flight
controls that are instantly accessible by both the student and the instructor (for
example dual flight controls or a centre control stick). Swing-over flight controls
should not be used.
(c) The fleet should include, as appropriate to the courses of training:
(1) aircraft suitably equipped to simulate instrument meteorological conditions (IMC)
and for the instrument flight training required. For flight training and testing for
the instrument rating, an adequate number of IFR-certificated aircraft should be
available;
(2) in the case of aeroplanes and sailplanes, aircraft suitable for demonstrating
stalling and spin avoidance;
(3) for the flight instructor (FI) training courses on aeroplanes and sailplanes,
aircraft suitable for spin recovery at the developed stage;
(4) in the case of helicopters, helicopters suitable for autorotation demonstration;
(5) in the case of a non-complex ATO, one aircraft fulfilling all the required
characteristics for a training aircraft might be sufficient;
(6) each FSTD should be equipped as required in the training specifications
concerning the course in which it is used.
FCL.013 includes a "review of AMC1 ORA.ATO.135 and possibly develop an additional GM to clarify what is meant by ‘adequate fleet of training aircraft’; to include training aircraft requirements for additional ratings under development;"
Subject to that AMC/GM, it will be up to individual NAAs to decide what aircraft can be approved for training under ORA.ATO.135. There is no prohibition on Annex II aircraft, nor is there an EASA requirement to permit their use.