I just thought of a possible other line of research.

Every person in the time would have been member of a church. If was Pollux was at its home port when the muster roll mentioning Hendrik Philip was drawn up, Hendrik Philip might also have had his home base in that town.

In the muster roll it is mentioned that Hendrik Philip is staying in the 'Raamstraat' with 'ant  Verver'. Looking at other entries of sailors that have been transferred to the Pollux, based on the addresses mentioned with them, I get the impression that Pollux was in port at Rotterdam. In the Rotterdam archives I found a reference to an 'Antonia Verver' living there in 1781.

So it might be an option to see if Hendrik Philip is mentioned as a church member of the Evangelical Lutherian church in Rotterdam. I have, for an ancestor of my wife, found reference to the home town of her ancestor in just such a members list of this church.

On familysearch.org there are several books of church members (lidmaten) of this evangelical lutherian church available: https://www.familysearch.org/search/catalog/122903?availability=Family%20History%20Library

On a completely different line. Looking at the name of Fasmer, I came across a transcription of a ship's journal from 1786 of a Dutch whaler to Greenland that had to get into Bergen after getting into difficulties. (link)
In this journal there is a list of crew members and among these there is mentioned a sailor (matroos) called "Jan Fredrik Carel" and it says he comes from the city of "Stettyn", or "Stettin" in current German writing.

He might be not related at all, but if you would want to take a stab in the dark, this might be a city to look at the baptism records, as the family name does occur there and they have become sailors.

Found something else that might be helpful

There exists a ships’s journal in the National Archives, of the frigate Pollux from the time Hendrik Philip was serving on this ship. If it was custom to mention all crew members and their origin in such a journal (like I found on the journal of the whaler) then it might offer something new:

162 Journaal gehouden door M. van Nierop, dienende als 1e luitenant op 's lands fregat Pollux onder commando van de luitenant-kolonel en kapitein W.O. Bloys van Treslong, 10 juni - 20 november 1795

(Hellevoetsluis, Texel)

 

Regarding the fate of the fregat Pollux, it is mentioned here (on the 4th page link ) that it has been sold in 1799.
Which is a bit conflict with the end of service of Hendrik Philip on May 12th 1800, when he was released from service from the muster roll of Pollux. Maybe he was on shore since the sale and not transferred to another ship?

I found the following {my translation} about the final years of the frigate Pollux (link)

... [Willem Otto] Bloys was in command of the frigate 'Pollux' from June 10th till November 20th 1795, but he never got much further than the Nort Sea and Texel. End of Februari 1796 he sailed with his ship in a squadron led by vice-admiral Adriaan Braak to the West-Indies to protect the Bataafse kolonial properties against the British. Mid May the squardon arrived in Surinam. Along the way they lost two of the eight ships. Once arrived Braak unexpectedly died quite sudden, after which Bloys on September 3th temporalily took over the command. In the meantime in Paramaribo they feared a blockade by the Royal Navy, that now and then showed itself in front of the Surinam Rivier.

Finaly on March 21st 1799, after a fruitless stay of two and a half year overseas, the squadron left Surinam, after which the colony was captured within 6 months by the British. Due to the dominion of the seas by the Royal Navy, they could no longer get back to the homeland and they ended up in Norwegian Trontheim, where the ships were eventualy sold for scrap value. On September 10th 1799 Bloys – since April 6th 1798 captain-at-sea – put down his command and returned to the homeland travelling over land.

...

You have given many good tips for further search here. I see (by simple googling) that there was a Raamstraat in Rotterdam at the time, so I will indeed work myself through the church records there. If he may be actually placed in Rotterdam, that could also be an indicator that I should go even more thoroughly through possible materials from the Admiralty of Rotteram/de Maze.

I also clearly need to find a way to consult that journal (no. 162). I was in fact aware of it, but hadn't seen other journals that also include a list of staff - normally they seem more like diaries to me - before that you linked to above concerning the whaler, so thought it was unlikely to contain anything of interest, but now I have some hopes.

Have a nice weekend,