What is the History and meaning of the symbol #23 (丸に七曜) in this image?
[Note: this became an answer because it was far too long to post as comments. It doesn't really answer the question, but will hopefully put OP or someone else on the right track.]
Some google image searching suggests it might be the heraldic symbol of one or more Daimyo clans:
https://www.patternz.jp/japanese-family-crest-list-symbol/ [see the Samurai-related section]
I failed to locate the symbol's precise meaning or origin, besides the obvious finding that it's the round of seven [days] symbol using Google translate. Later googling further corroborated what Google translate suggested, with small variations but around the same theme.
That being said, that last site I linked to (patternz.jp; no affiliations) apparently offers a book on the subject, and they seem to sell a service to look it for you.
As to which families might be using the symbol or very similar ones, I found the Tanuma and the Kuki families, among others:
http://www.asahi-net.or.jp/~me4k-skri/han/toukai/sagara.html [JP; no idea how reliable this source is; it lists other families on other pages with a similar symbol, e.g. Matsudaira]
http://www.diffworlds.com/samurai/Provinces/totomi.htm [fiction site, but that specific page appears to be based on reference materials]