You might do very well with the Shurley jingles...one of the criticisms that I recall was that various jingles have similar lines in them ("easy as can be" comes to mind) which can cause you to jump track and end up in the wrong jingle. This sort of thing happens often here in, for example, Prima Latina prayers -- a single Latin word that is common to several prayers can totally derail my younger dd. It sounds as though your dd does a better job on memorization.
And, one gripe I have about R&S is the lack of jingles. They just state "memorize the forms of to be". Fortunately we knew the chant from FLL -- am(clap) is(clap) are,was,were(clap) be(clap) being(clap) been(clap clap). So, R&S is sort of dry from that viewpoint. Sometimes I add in some Montessori or Waldorf stuff to give it pizzazz. Also, they call statements "telling sentences", which annoys me for some reason. On the plus side we are learning to diagram sentences, and there's enough composition to ditch Writing Strands -- not that we ever DID Writing Strands, but now I don't feel guilty about skipping it.
The religious aspect of R&S is okay if you're Christian (I've heard their history is offensive to Catholics, don't know about the English -- we're Protestant, so I don't tend to notice those things). They use Bible stories to illustrate various points, like "find the simple subject and simple predicate in these sentences" followed by 10 sentences telling about Noah's ark or somesuch. They don't proselytize about the stuff, though, any more than a secular text would try to convert you to belief in fairy tales or Charlotte's Web. These are simply stories they use because they are common in the culture for which the texts are written...EVERYBODY (Mennonite/Christian) knows Paul and Silas sang in a prison, right? No need to explain why or when they sang or why it's important to YOU that they did this -- it's just a story everybody knows, so they use it. The nonBiblical parts reflect Mennonite culture -- big families, farm life, girls clean and cook, boys help Dad, missionary trips, church attendance, the people in the pictures all wear conservative Mennonite clothing (long sleeves, long pants for boys, longish dresses for girls, headcoverings for women). We were at the zoo last week and saw a family that looked as though they just popped out of the pages of R&S, which dd thought was very cool.
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