No you leave the starter connected until after the car starts.
A few issues with the lithium packs, they need to be charged carefully at set voltages
and currents. I wonder how long they will last when the car starts and sticks out
over 14 volts onto the 11.1v pack and upwards of 60 amps.
Would you want to leave one in a hot or cold car?
Car batteries are designed for high current outputs for a short duration then to be
fully charged as you drive. If the battery is less than 100% for any length of time the
the plates sulphate which coats them and reduces the capacity. Continue to allow
the battery to stay in a less than fully charged state will ruin it.
Spend the money on a decent battery to start with and a decent charger afterwards
and keep it topped up.
I think you said initially it would hold its charge for weeks? If your not doing longer
journeys to charge it then use an external charger. Dont let the charge drop.
PS.. The quoted max power on those devices are often miles off the mark, 300Amps
peak power maybe for a tiny fraction of a second. What is the continuous cold cranking amps?
I was contemplating a couple myself but some yourtube testers put me off, ove claimed 800amps when struggled to put out over 200amps. Several expanded and broke their own cases under load. They are a fire hazard when that happens.