The compounding of amiodarone oral solution from tablets generally involves adding methylcellulose or adding in a certain amount of 8.4% sodium bicarbonate (same as the injection) to balance the pH to around 7. This will likely solve the precipitation issues. You would have to add in enough to reach this balanced pH (testing with litmus paper or probe).
The PHYSICAL stability of these formulations are found in multiple studies as 90 days refrigerated. However the general USP 795 recommendations for compounded non-sterile aqueous formulations is limited to 14 days refrigerated, so set yours to match especially if giving to NICU patients. USP 795 accounts for the risks of MICROBIAL growth in making the BUD limit of 14 days even though the PHYSICAL stability can be longer.
Attached are compounding formulas for 5 different children’s hospitals that essentially say the same thing. Two do not say how much sodium bicarb to add but 2 others do, and it is about 4 mL with each three 200 mg tablets according to one formula, and 12 mL with each 5 tablets to another, so you could take an average of the two and work with that if you do not have any means to test pH.
If you have found the tablets to dissolve in simple syrup alone this could have a slightly different pH than what has been studied and may have shorter physical stability making it very hard to determine a uniform stability time for your batched product. Inspect the product each time you dispense to ensure it has not started to precipitate. I still think the physical stability would be longer than that USP 795 stability of 14 days so keep it at max 14 days refrigerated and you should be ok. Each of these formulas have the literature references you would be looking for.