The kind of "can run on a desktop computer" handwaving info suggests that this is an electret condenser capsule rather than a "true" unpolarised condenser microphone in need of phantom power.
The phantom power, if any, will likely only be used for the internal FET preamp instead of polarizing the membrane. While the microphone is apparently fine with 48V, it does not need it. Instead it can get along with "plugin power" supplied through its 3.5mm cable. It's also unlikely that the connection is balanced even when using a balanced XLR cable (rather than the delivered one) and an actual microphone preamp with phantom power.
That means you won't get the common noise rejection of a balanced connection, and the quality of your phantom power circuit might depend on it. Also, you'll likely become much more susceptible to the switching power problems of your laptop supply by having another power supply in the loop without using balanced connections. This is a classical ground loop problem.
Try working without the phantom power supply, using "plugin power" instead (it's usually available on most mic inputs of soundcards or laptops, possibly by using some utility for switching it on).
The microphone you are using is not as much budget equipment as cheap equipment. I am surprised they even specify an equivalent noise level in their data sheet, and it is not actually bad for a small condenser capsule of 16mm diameter. Too good to be true would be my guess. Maybe they have just reprinted the capsule's spec without bothering to measure the whole microphone including delivered cable and their own plugin power circuit. At worst they have copied the specs from an unrelated microphone.
Personally, I don't think separate phantom power supplies make a lot of sense for computer recording these days. External sound cards of some quality will readily supply phantom power and you don't have to meddle with batteries or ground-loop-susceptible additional power supplies.
You'll still be able to use your BM800 with one of those and it will likely get the best out of that mic that you can hope to get, but considering the price of, say, a Samson C02 pair of microphones (still on the cheap and thus noisier side of budget, but manageable with good placement), meddling with plugin power designed equipment and unbalanced cables is just not worth it.