Coir and perlite or pumice, and a little activated charcoal to balance fertilizer and pH. A little bit of sphagnum moss near the bottom roots provides some extra protection from dry indoor air.
No compost or smelly organic fertilizers. The outdoor microbes will die indoors anyway, so you end up with a fly problem once it starts leeching compost juices.
Liquid fertilizer and slow release Osmocote pellets are best, in low doses, or none when the tree is transitioning or sick.
You can try lime again. I know mine died a few years ago because the soil was too much wet organic matter (peat). A hardy variety might survive better year after year. Persian Lime I think is what local nurseries sell for our climate.
Store bought cactus/citrus soil has worked well in my experience in larger scales. It's good, really available, and serves as a nice low maintenance soil for periods when it just needs to leaf out. Not that heavy. Generalist stuff.
I will apply your tips in my next generation of bonsai clones. Small amounts of the big bag soil really fluctuates unpredictably in terms of moisture content.
Yeah that sounds like it will work. Rooted cuttings probably need no fertilizer at all. Just keep it covered with a cup or humidity dome if you can. Same with a graft, keep it tightly wrapped in plastic.