GG5: hot sake with beaten egg in it; a sake 「eggnog [flip].
ルミナス: a hot drink consisting of sake, egg and sugar

I agree "eggnog" is a bit off the mark.
I don't think the romanized version is common enough to be a useful gloss. Better to keep it simple and descriptive..

@@ -18,3 +18 @@
-<gloss>tamagozake</gloss>
-<gloss g_type="lit">egg sake</gloss>
-<gloss g_type="expl">hot drink make with sake, sugar, and raw egg, sometime used as a cold remedy</gloss>
+<gloss>hot sake with beaten egg and sugar</gloss>

https://en.wikipedia.org/wiki/Tamagozake

Three hits on reverso, all tamagozake (but all one reference I think)
https://context.reverso.net/translation/japanese-english/卵酒
For examples in which warmed Japanese sake is used for ingredient of cocktail, there are Orange Sakini and tamagozake (egg sake).

https://vintageamericancocktails.com/tamagozake/
What Does The Tamagozake Taste Like?
The Tamagozake is a Japanese cold remedy drink like how the hot toddy is in the United States.

https://www.hirokoliston.com/egg-sake/
It’s a Japanese version of Eggnog, but it got to be hot. This alcoholic beverage is called ‘Tamago-zake’ in Japanese, and it is offered to anyone who have a cold, even children. My mother always made it for me when I had a cold

"eggnog" is one of these cultural "glosses of convenience".  eggnog contains dairy (cream).  tamagozake is not eggnog.

"egg sake" appears in English references, but usually with tamagozake nearby.

The "cold remedy" part is useful/important for interpreting Japanese literature.  The cold-remedy aspect of the drink comes up frequently in English-language references.(some refs included)

Note: "Japanese eggnog" doesn't work as a gloss.  If you google this, you'll get recipes for actual eggnog using Japanese alcohol.

I'd consider cramming "eggnog-like" in the [expl], but I don't actually think a beverage without cream is eggnog-like at all.  Eggnog is usually served cold, is often alcohol free, is loaded with cream (&fat).  You can't have alcohol-free tamagozake....

@@ -18 +18,3 @@
-<gloss>eggnog (made with sake)</gloss>
+<gloss>tamagozake</gloss>
+<gloss g_type="lit">egg sake</gloss>
+<gloss g_type="expl">hot drink make with sake, sugar, and raw egg, sometime used as a cold remedy</gloss>

N-grams:
卵酒	9055
たまご酒	7756
玉子酒	6363

@@ -18 +18 @@
-<gloss>eggnog</gloss>
+<gloss>eggnog (made with sake)</gloss>

daijr

@@ -8,0 +9,3 @@
+</k_ele>
+<k_ele>
+<keb>玉子酒</keb>