Perhaps the unit that you are using is similar to that shown in this link:
https://www.google.com/url?sa=t&sou...FjAAegQIBBAB&usg=AOvVaw3veZLI5jQkbCwYz5Lc0X3M
The instructions indicate to use 2 boxes (3-4 lbs) of regular table salt placed into the pre-filter sump, which appears to be similar to the procedure you are currently following. As the solar salt will likely dissolve slower than table salt, that maybe why you are considering pre-dissolving salt to create brine which would take-place in the additional bucket.
Perhaps a simple diagphram pump could be utilized to transfer the brine from the bucket to the softener outlet connection. Here is an example:
https://www.amazon.com/OMMO-Diaphragm-Priming-Sprayer-Pressure/dp/B0863WL6FG
Regeneration would then involve adding the appropriate quantity of salt and water to the bucket 1-2 hours in advance of regeneration. When brining is to occur, the pump will then transfer the brine from the bucket to the softener. Once the full strength brine has been transferred, additional plain water could be added to the bucket, allowing the pump to further slowly rinse the calcium, magnesium, chloride and excess sodium from the resin bed.