The idea will probably work, but I would recommend something other than any product from Efest or similar places. :/
Some mentions just here on ES:
search.php?keywords=efest&terms=all&aut ... mit=Search
Keep in mind that if the paralleled unprotected pack does not actually connect thru the BMS of the protected pack, that BMS cannot protect the paralleled pack at all--it will not shut off current from it, it will only shut off current from the pack the BMS is already connected to protect.
If it is possible to parallel the new pack to the existing one at the cell level, then you dont' need any extra charger/etc; the BMS and charger of the existing one will do all the work for you.
As far as the balance charger goes...as long as you never "cross the streams", by connecting any wires across series pack connections, you can balance any paralleled cells in whatever way the chargers work.
If you have a 6s charger, you'd connect the most negative 6 parallel groups to the charger, then disconnect those when full. Then connect the most positive 4 parallel groups to the charger, then disconnect those when full.
There are 10s balance chargers, too, though you may need to split the pack in sections, breaking series connections, with some balance chargers. IIRC, some of the 10s chargers do not charge 10s packs; they charge two 5s packs that are not in series with each other. Make sure the charger you intend to use does not require this.