"strollingly" is not really an English word. I think the other glosses cover it.

@@ -14 +13,0 @@
-<gloss>strollingly</gloss>

散歩がてら行こうか？ is hard to translate but the meaning is basically "do you want to combine going there with a bit of a stroll?"
I feel this nuance isn't properly conveyed by "while out on a walk" alone.

"strollingly" is a bit awkard but kind of manages to convey what I'm referring to?

@@ -12 +12,4 @@
-<gloss>while taking a walk</gloss>
+<gloss>while out on a walk</gloss>
+<gloss>while strolling</gloss>
+<gloss>strollingly</gloss>
+<gloss>in a strolling manner</gloss>

I won't object. It's in a Tanaka sentence, but ut's our only example of がてら.

散歩がてら	204706
運動がてら	14181
行きがてら	14584
花見がてら	9399

daijr, meikyo, koj all have 散歩がてら in their example sentences for がてら
(the other ngrams are from other examples in daijs, nikk, etc.)

gg5 has 散歩がてらにお寄りしました.　I have just dropped in while taking a walk. (散歩 entry)

39 reverso sentences
https://context.reverso.net/translation/japanese-english/散歩がてら

eij sentence
散歩がてらホテルへ戻る
stroll back to the hotel

prob most common -がてら construction, maybe worth having?