welcome to stack exchange!
First, let's think about your perspective on this problem. It seems odd or difficult if you think about it in terms of using this high impedance ground connection as a power ground. But that's not what it is for.
This is actually very typical among electronics, something called 'open-drain/open-collector' logic. This is essentially just a NPN or NMOS transistor that connects or disconnects ground through its drain or collector (which is open or unconnected if nothing is connected to the pin). Typically there is a series resistor for current limiting, though not always. Regardless, I would not continue drawing 150mA from the connection, it is probably not actually rated for that much current.
The line is a logic line, it is not meant to provide meaningful current or power, but rather simply control something else.
Fortunately, it is extremely simple to interface with open drain outputs. These kind of outputs I like to imagine like those cords that you used to be able to pull on in a bus to signal it to stop.
And indeed, we must use a pull-up resistor to interface with this. And there are few different ways to accomplish this, but the simplest is to simply use a PMOS/P-channel MOSFET. And everything is actually going to be in reverse of how you're thinking.
Your device should always be grounded and always have a solid, always connected connection to ground. This is the standard convention for automotive designs, and indeed, check any exposed metal anywhere in your car, and you'll see that it is connected to a common ground. So you should not switch the devices ground at all.
However, don't connect the +12V. It will instead connect to the drain of the P-channel mosfet. The source of the mosfet is then connected +12V.
MOSFETs are off when their gate is at the same potential as their source. So we connect the PMOS's gate to +12V but through a pull-up resistor. Ideally, it should be 10K-100K, it must be a good deal high impedance than our control line.
This will ensure the MOSFET, and your device, are normally turned off. In this state, the +12V is effectively disconnected from your device.
However, if you connect your open-drain control line to the gate as well, when it is 'on' (connecting ground though the 1K resistor), it can pull the PMOS's gate down since it can over power the higher impedance trying to pull the gate up. And to turn a PMOS transistor on, you need (usually) a potential 10V lower than its source. So like a NMOS transistor, but in reverse (which needs 10V above its source).
All roads lead to Rome however, and this is but one road. There are a dozen other ways to accomplish this, this is just the first way that came to mind. Though many alternatives will involve actually disconnecting the ground, which I do not recommend in an automotive environment.