Not a teacher like mrz and feenie but my ds sounds similar to yours.
We got bear necessities with my ds in year 1 as he was getting confused by his phonics and look and say reading books sent home. He has a similar birthday to your ds ,also 1b, is on blue level books at the moment and I have found dancing bears good for checking what he knows and what he doesn't and reinforcing ,for instance he was confident on oa but not oar from the 1st book.
The end of the 1st book is should,would etc. The programme checks for 100% confidence and fluency, so the 'lack of sticking' that you were concerned about. Yes, you do cover things he will already know but otherwise there can be gaps in his knowledge that you don't realise.Talking to teachers who help older children who are struggling, often this is how they start.
You could start on book B includes -tch,-dge,-all,-ge,-ce with lots of repitition which I feel helps. There are several stories in the workbooks! but not the most exciting reads.
I would still borrow library books, but you may struggle to find enough phonic readers as alot of the books are just like the old oxford reading tree with a mix of phonics and look and say words.
You would probably need to look up some phonic specific reading books and order them in unless your library is particularly big and well stocked and this is where the reading chest comes in I suppose as it saves the hassle.
Red books don't seem to equal Dr Suess level of reading,so you need to ask the teacher what specific things are stopping her putting up a level. With my ds it is lack of fluency, as his decoding skills are apparently very good and he could be on a higher level if he read more fluently. I wouldn't be happy if my ds was still on red level from what you say you already do at home with him,which is more than alot already. It is very odd he is 1b,which he is where he should be but is on reading books that would be below this.