The delay is probably nothing to do with the seller.
I have never had any problems ordering from China.
I no longer buy anything from the US, far too expensive.
I was testing my SX220hs cameras yesterday with the external Pololu effectively feeding the battery contacts .
The Pololu is set at 4.8V and falls 'a bit' under load.
I forgot to remove the camera battery.
For some reason it did not cause any harm, the battery and converter both work fine.
So long as the battery has a built in protection circuit (assuming it is a LiPo), and assuming connecting the converter to your setup doesn't disconnect the battery, then all that is going to happen, is that the regulator will charge up the battery to its maximum voltage, around 4.2V and the protection circuit will kick in, in a manner similar to the way that little solar charger works. If the battery was unprotected (very unlikely) then I wouldn't stand too close
In the case of the SX220HS, the battery is LiPo and is protected, so you are pretty safe, you just end up charging the battery, which probably adds about 0.5A to the load on the regulator until the battery charges completely.
If the battery in question was an alkaline pack, it would also probably charge (and eventually overheat and leak), it is possible to recharge both alkaline and zinc carbon cells with a specialist charger, albeit they usually have reduced capacity and can only be recharged a limited number of times, so a one off event like this would probably not be catastrophic. Dont get me wrong, this is not a recommended idea, but if there were no fizzes, pops or bangs, then everything is probably unharmed.
Those little solar chargers actually connect the solar panel directly to the battery through a schottky diode, so far as I can see there is no path through the regulator as I originally thought, so they only actually charge the battery when the solar panel output voltage, less the diode drop exceeds the current battery voltage. Its a bit crude, and probably woefully inefficient, but it seems to work.
I flattened the black charger completely and stuck it in the sun, to see if the solar panel was just an ornament. The built in, nominally 0.4W panel does actually charge the battery, albeit pretty slowly.
I was meaning to do the same thing with the hacked about gold one, to see how long a full days charge in the sun actually lasts, but other thing got in the way,