I agree, new? Call PM and ask for a replacement.
If is is an LED unit, it is unlikely that there is a bulb that can simply be replaced, but who knows... .
Can you simply remove a bulb? If so maybe post a photo? You indicated 27V at pin 1 of the socket? Maybe 24vac? Measure the pin spacing and size etc... and then you can find lots of available LED bulbs that may fit. Read on, I did a little searching for you.
The lamp is labled as JC34A and the voltage at the 2 pin socket is 27
You said it was and LED LAMP Bulb. I looked at the PM photos of your lathe. I also googled the part number you gave, which I do not think is a "standard" part number for a bulb? I also found it at JET tools. (Machine Lamp Size Jc34A | GHB-EL) From this effort, I assume the part number is for an entire lamp fixture not just the "bulb" as I found little in the way of hits for just a bulb. But did find some leads.
https://integritycrane.com/product/machine-lamp-size-jc34a/ which is priced (~$100) like an entire lamp fixture including integrated LED.
https://jettools.com/ghb-1340a-electrical-schematic This shows it wired to 24ac.
A German site said it was a 40W 24V part, but little else.
Your manual says it is 24Vac, page 35, schematic plus the paragraph at the lower left side.
By the way, there was virtually no info for the light that was supplied with my PM1440GT. However, there was label which indicated that it would run on either 24Vdc or on 110ac? In your case I think the JET schematic shows it on 24Vdc so I would stick with what ever it is wired up to at the lathe supply! The modern LED lamps and housings are full of electronics. They are not just simply and LED with a resistor in series like an indicator light/LED. The put out considerable power so ... There are all kinds of ways to wire up LED circuits, especially when there are multiple LED junctions involved. Anyway, sometimes and LED is a bunch of LEDs wired up together inside a glass bulb shape and some times they look entirely different.
I have installed a bunch of dimmable LED lamp fixtures in my home. They run on 110Vac, but if you look at the integrated circuit that is in the fixture you find that the LED junctions are running at ~ 2.5-3 volts DC and that the circuits use pulse width modulation at high frequency to charge a capacitor circuit that then bleeds into the raw LED (junction). This is ironic as the classic wall switch dimmer also uses pulse width modulation to modulate how long the 60 cycle line is supplying current. The dimmer switch does not reverse the polarity with the 60 cycle. However, the LED modulator circuit does use both halves of the power cycle by effectively reversing the leads at the 60 cycle zero crossings. Rather complex, but efficient. It is all on a chip packaged with the LED junction(s). It is great what you can do with modern electronics if they do not fail!
By the way, on my mill, I found that sometimes the hot cutting debris was so hot that it stuck (embedded into) to the lamp glass cover/lens. I had a heck of a time cleaning it and when done it was still pretty beat up. I was even using flood cooling at the time and this was also splashing on it. So, I put a plastic sandwich baggie over it and while the bag looks like heck it is disposable and it deflects the hot cutting debris ... somewhat.
Good luck
Dave L.