AFAIK Avaris in the Delta was the all seasons capital. You have listed 4 of the 6 the kings of the XVe dyn (the number varies from one egyptologist to the next). Their official title was heqa khasout (ie. ruler over hilled countries)which is also know as Hyksos. During the SIP in Egypt, all sovereigns (Kush, Thebes) were an heqa. As far as Egyptiens of the NK and the TIP are concerned, they never called a group of people : Hyksos.
I realize Barlier is no longer part of this group, but some of the points raised in this thread may be significant.
True, the term "Hyksos" is not used in Egyptian texts. These kings did not include Heqa-Khaswt in their titulary, neither did contemporary Egyptian pharaohs refer to them by that name.
We have Josephus/Manetho to thank for the erroneous assumption that the term Hyksos refers to an entire people. It did not, and it is totally false. However, for our use it is a very convenient term to use.
The title Heqa-Khaswt does appear on several scarabs of these 'foreign' kings, but what does this term really mean?
If I assume the title "King of.......", you can rest assured I want you to know what I am king of, who am I ruler over.
"Ruler of Foreign Countries" tells the reader nothing, as a title it is next to useless, except that it informs the reader that I am a Ruler. But what am I ruler of; Canaan, Naharyn, Ashkelon? - why not tell the reader where I am ruler of?
If I am just the ruler of some unnamed expanse of wilderness, way over the horizon, then where is the prestige in saying that?
And, why do several kings all make the same claim?
I think we have missed the true meaning of this term, Heqa-Khaswt.
The "tripple-hill" determinative certainly reads "Khaswt", and depending on the context can mean; land or country, but it does not mean "foreign".
The determinative for "foreign" is the "throwstick", and no "throwstick" is used in this title.
There are many examples of the "tripple-hill" determinative used for lands within Egypt, typically lands, perhaps suburbs, of great cities. These lands are widely known, and obviously they are not foreign - and, no "throwstick" is used in those terms either.
So, the "tripple-hill" determinative (Khaswt), means either 'land' or 'country', but does not automatically assume a meaning of "foreign", unless the "throwstick" is used.
This being the case, the title these 'Hyksos' used merely advise that they are Rulers of Lands, but which lands?
Interestingly, Apopi, the ruler of Avaris, does not use the term, Heqa-Khaswt. Yet we commonly assume he was a 'Hyksos' king.
That said, he does adopt a similar title - Heqa-Het'waret. Which some of you may recognise as meaning, "Ruler of Avaris", which he is known to have been.
Scholars have known this for a long time, yet they do not seem to have recognised that if "Het'waret" refers to a defined place, then possibly "Khaswt" also refers to a definite place.
If that is true then the title Heqa-Khaswt must identify a specific place which may have been lost, or gone through a name-change, down the centuries.
Was there a place in Egypt called Khaswt?
Yes, this was the ancient name for Xois. The name of the 6th Nome of Lower Egypt, and its capital city was at one time Khaset/Khaswt.
Heqa-Khaswt must mean Ruler of Khaswt, the 6th Nome of Lower Egypt.