It sounds like your device needs more than the 100 mA it is guaranteed, but does not enumerated properly to request the extra current. You can get up to 500 mA from a USB port if you ask for it and the host grants it.

There is no guarantee that the host will grant the request, but desktop machines pretty much always do since they have plenty of power available. In fact, the desktop motherboard schematics I have seen show only a polyfuse between the 5V supply and each USB port. There is no active electronics limiting current. On those machines, you get the full 500 mA even if you don't ask for it.

Laptops however have limited power and therefore actively monitor and control USB power in most cases. Some machines may limit the sum of all USB devices to a certain current, or limit it more when running on battery only, or when the battery is low, etc. Note that USB powered hubs can't ever grant the full 500 mA since that is all they are getting themselves.

The reason your device won't enumerate anymore is because you broke the ground connection. All three lines D+, D-, and ground are required for communication. You can cut the power line if the device is self-powered. Self powered devices generally have no connection to the USB power line. That's perfectly OK, but they still have to connect via ground.

The problems started when the USB connector of the original device got damaged. Would it be reasonable to repairing the connector? You could also try to bypass the original connector to feed the charger.

Separate charger approach

The proposed approach with separate charger is sound. However, I would look for a charger, with:

• A bit less current. Lithium-ion cells should be charged at current of 0.2C to 1.0C. This means that capacity of the cell drives the charge current and charge time. 0.2C is a shorthand which means that the current in [Amperes] is 0.2 of capacity in [Ampere-hours]. Units don't quite match, but again, this is a shorthand. 1A is on the upper side of the range. I would look for something between 0.2A and 0.6A.
• Published schematic or known part number for the charge controller IC. This improves the chances during troubleshooting.

At the same time, there's nothing fundamentally wrong with the charger, which you've found on eBay.

The battery pack connector looks like Molex PicoBlade. Here's a 4-pin male (p/n 53047-0410).