Its very confusing without drawings but...
I think if you do the PD first then the PP work, then your PP works would not comply with your approval.
Your PP approval will cite the works must comply with drawings which will include as existing. Once you have done PD works the existing is then not correct, and therefore you cannot claim to be complying with the works. Furthermore/also at no point would your development have complied with the proposed drawings either so it is hard to say you have complied with either the existing or proposed drawings should this be challenged.
Whilst you are correct the PP does not strip PD rights, this is only in that it doesnt say so in writing because it does not need to! In practice it almost always stops PD extensions that would be connected to an existing extension, and especially those existing extensions that are gained under PP because of how the TCPO (GPD) 2015 Order is worded - basically the combined/connected extension would have to meet the PD rules (and if that's the case then there is no reason why the existing extension would have sought PP rather than just being done under PD.
This is perhaps where the thought doing the PD works first comes from, but as above I don't think the PP is then usable anymore, and instead a new PP should be sought.
The above is also explained/shown in the PD technical guidance:
any total enlargement (being the enlarged part together with any existing enlargement of the original dwellinghouse to which it will be joined) exceeds or would exceed the limits set out in sub-paragraphs (e) to (j); Where the proposed extension is to be joined to an existing extension to the original house, whether that was built following a planning application or under permitted development rights, the total enlargement (being the proposed extension together with the previous extension) must meet the limits set out in (e) to (j) above.
The example the guidance shows is adding a first floor extension on top of an existing 4m single storey extension, which makes the whole extension not compliant and therefore the first floor cannot be done under PD as geraldthehamster said.
It is worth noting and being cautious as the example cited is from 2002 (and uses the 20 year old GPDO from 1995) and also involves distinct/separate works, whilst the two GPDOs are similar they shouldn't be seen to be interchangeable, and in fact the GPDO from 1995 also had many amendments to and I'm not clear what form it was in 2002. The website
http://www.permitteddevelopment.org/ does not make clear what the advice is based on and is ran by a Technologist who may have knowledge in this field but usually their expertise would not be complicated Planning Law.
My thought would be without a recent case law example based on the current GPDO would to be cautious in carrying out extensive and expensive works.
Another thing to note if you put in an application showing PD works that are not yet started (but say with a a label referring to a gained LDC) the LA will request drawings without the PD/LDC works shown. You could still show them and refer to the LDC but you are effectively asking for consent to all the works in one go.