Hello and thankyou for using this service!
I am sorry to hear about the problem you are having with your foundation
Sounds like you are right on track with the way you want to go with it.
The only problem I would have with it would be using lag bolts. You should clamp your scab to the broken joist and drill thru both and use a carriage bolt with a washer on each side.
Double up on the nuts and tighten them against each other to lock them in place.
The lag bolts can become loose over a period of time. The bolt will not.
When you are ready to tighten them up glue them together and assemble.
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