You can definitely do this (I've done it many times). I'd start by selecting a 12 VDC LED bulb instead of the incandescent bulb you showed. An LED bulb will generate less heat, and can use a smaller power adapter since they use much less power than incandescent bulbs.

There a "billions" of 12VDC LED bulbs on amazon. Here are some considerations:

- Base/socket type: these bulbs last a long time, so I typically don't use a socket. I select a bulb with pins (instead of a bayonet base) that allows me to directly solder wires to it. But using a socket is fine, too; it just takes more space.

- Dimmable: Not all of this type of bulb are dimmable (see more about this later)

- Color temperature: These bulbs are available in cool/neutral/warm white

- Light beam pattern: There are many different designs/shapes that distribute light differently. Some are very directional (unlike an incandescent bulb), so pick one that works well with the fixture design.

Once you select the LED bulbs, you'll know how much power is required (in Watts). Pick a power adapter than can provide at least 20-25% more power than required. This improves reliability/lifetime. You can easily find 110/220VAC -> 12VDC power adapters that have wires on both ends (instead of a wall plug). You can use Wago lever-nut connectors to make the AC-side connections. You could use these on the DC output side also, but I usually solder those connections and cover with heat shrink tubing.

There are two ways to dim the lights:

1) Use a normal 110/220V wall dimmer switch in the circuit feeding the lamp. Unfortunately, only a specific type of 12VDC adapter is compatible with this approach, and they tend to be expensive.

2) Insert a wireless dimmer into the connection from the power adapter to the LEDs. This works well (with dimmable bulbs), but the remote control unit is a bit cheesy and easy to lose.

Feel free to ask follow-up questions ;-)