Elizabeth is expecting her second baby -- a brother for son Bertram "Bertie".
Elizabeth writes: "I was utterly convinced that this time we were having a girl, to the point where I have only made a list of girls names and since having a scan and finding out we're expecting another little boy my husband and I have been in complete disagreement with each other's suggestions.
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I absolutely love "Sunday Best" names with quirky nicknames! It was a top criteria for my partner when naming our son, so I can relate to your criteria. The hardest part when compiling this list was the no diminutive ending in -i.e. criteria. It's just such a prevalent hypochoristic ending!
Alistair - This distinguished Scottish form of regal Alexander has plenty of dashing charm with the breezy nickname Alick. It ranks at #459 in England and Wales (E&W) but is currently on a plateau, meaning it unlikely to rise any time soon.
Barnabas – The Greek form of the Aramaic name meaning "son of encouragement" or "son of prophecy" The Anglicised form Barnaby ranks at #259 but Barnabas is #1001 in E&W. Baz could be used as a nickname.
Caspian / Cassian – Cassian is a stately Roman name, while Caspian is a literary choice. The former ranks at #731 and the latter at #768 in E&W. Potential nicknames include Cass, Caz and Cap.
Constantine – Meaning "constant, steadfast", this royal name has a fits between Cosmo and Augustus on your list. Stan could be used as a short form.
Edmund / Edwin – Ranking at #622 and #493 respectively, these two charming Anglo-Saxon royal names share Bertram's Germanic dithematic ancient heritage. Ed, Ned or Ted lend themselves well as diminutives.
Everett / Everard – Everett is the later (and surname form) of Everard, descended from the Old Germanic name Eburhard "brave as a boar" which matches Bertram's Germanic "bright raven" perfectly. Everett currently ranks at #1369 in E&W; Everard is unranked. Vere or Rhett (for Everett) could be used as nicknames.
Fintan / Finnegan – Lending themselves to the friendly Finn as a nickname, Fintan (meaning "white, blessed fire") and Finnegan ("little blessed, fair one") are both options.
Gus – Starting backwards from the diminutive, I noticed Gus made your list. If Augustus is off the table how about Angus, August, Augustine, Fergus or Magnus as a way to get to Gus
Ignatius – Stately Roman Ignatius is another option for Gus, or even Nash or Nate. It currently ranks at #2728.
Kester – A Tudor vernacular form of Christopher with a literary heritage. It ranks at #2728 and can be shortened to Kit or Kes.
Leander / Lysander – Two strong, heroic Greek names meaning "lion of men" and "freer of men" respectively which can be shortened to Leo, Lander, Anders etc. Leander ranks at #1745 and Lysander at #2266 in E&W.
Leopold – A princely name which is also a Germanic dithematic name (like Bertram) meaning "bold (man of) the people". It ranks at #974 in E&W.
Llewelyn – A handsome Welsh name, borne by royalty, which means either "lion likeness" or "leader likeness" that can be shortened to Lew. It ranks at #1316 in E&W. The medieval Anglicised form is Leolin / Leoline which is currently unranked.
Malachy – An Anglicised form of either Máel Sechnaill -- "disciple of St Sechnall" (ultimately derived from Latin Secundus "second") -- or Máel Máedóc -- "disciple of St Madoc" ((ultimately derived from Aed "fire") which ranks at #1113 and can be shortened to Mal or Mac.
Peregrine – Unranked in 2019, though ranking at #3412 in the previous year, Peregrine is a Latin name meaning "traveller". It is also the name of of a falcon -- matching the avian meaning of Bertram. Pip could be used as a diminutive.
Quentin / Quinton – Both deriving from the Roman name Quintus, Quentin ranks at #952 and Quinton is at #1113.
Ranulph – Ultimately derived from the dithematic Germanic name Randulf ("shield rim" + "wolf"), Ranulph ranks at #2728. Ralph or Rafe could be used as short forms.
Raymond – A Germanic name, established by the Normans which can be shortened to sparky Ray. It ranks at #597 in E&W.
Roscoe – A breezy Viking-heritage name which has had popularity in America but is underused over here at #798. Ross, Roe and Rocco could be used as nicknames.
Sylvester – From the Roman name Silvester meaning "forest" and ranking at #1869 in E&W. Sly or Sev are nickname options.
Thaddeus – Often shortened to Tad (though you could also use Ted), Thaddeus is an Biblical name with various meanings.It ranks at #2266 in E&W.
Zebedee – A sweet and breezy Biblical name -- not unlike popular Zachary -- which means "gift of God". It ranks at #2266 in E&W and can be shortened to Zeb.
UPDATE: Welcome, Barnabas Nathaniel!