G: TL;DR:
This is very doable and already solved. You don’t need to build anything—just plug a 30-pin Bluetooth receiver into the dock. Your lamp becomes a Bluetooth speaker immediately.

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What you actually have (important)

Your lamp is basically:

• Speaker + amp
• With a 30-pin iPod dock acting as the audio input

That dock already outputs line-level audio (not digital-only), which is why this works cleanly. citeturn0search6

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The simplest working solution (plug-and-play)

What you buy

A “30-pin Bluetooth audio receiver”

image_group{"query": ["30 pin bluetooth adapter iPod dock small dongle", "iPod dock bluetooth receiver plugged into speaker dock", "30 pin bluetooth receiver adapter close up", "iHome iPod dock bluetooth adapter installed"], "aspect_ratio": "1:1"}

What it does

• Plugs directly into the dock
• Pretends to be an iPod
• Receives Bluetooth audio from your phone
• Sends audio into the lamp speakers

These are literally made to “convert any iPod dock into a Bluetooth speaker” citeturn0search5

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How to set it up (takes ~30 seconds)

1. Plug adapter into the dock
2. Turn on the lamp/speakers
3. Open Bluetooth on your phone
4. Pair to the adapter
5. Play audio

Done.

Most are plug-and-play with ~10m range citeturn0search0

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Why this works (important distinction)

The 30-pin dock carries:

• Power
• Control signals
• Analog audio output

So the adapter doesn’t need to “hack” anything—it just injects audio like an iPod would.

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Failure cases (where this might NOT work)

This is where your situation could break:

1. “Authentication” docks (rare but real)

Some docks expect:

• A real iPod handshake
• Apple authentication chip

Result:
Adapter plugs in → no sound

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2. Charging-only docks

If your lamp:

• Only charges iPods
• Uses AUX elsewhere internally

Then the dock might not expose audio lines.

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3. Weird proprietary control logic

Some combo devices (lamp + alarm clock + speakers):

• Route audio through firmware logic
• Require “Play” signal from iPod

Adapters usually handle this, but not always.

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If it doesn’t work (Plan B)

Option A — bypass the dock entirely

Open the unit and:

• Tap into the internal speaker input wires
• Add a $10 Bluetooth receiver board

This is 100% reliable, but requires:

• Soldering
• Opening the device

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Option B — use AUX (if it exists)

Some of these lamps secretly have:

• 3.5mm input on back/side

Then just use a generic Bluetooth receiver.

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What about using the iPod itself?

Not worth it.

You could:

• Hack a Shuffle → Bluetooth transmitter
• Or use headphone jack + transmitter

But that turns into:

• Battery management problems
• Extra devices
• Worse audio chain

The adapter skips all that.

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Bottom line

• Feasible: Yes
• Effort: Minimal
• Cost: ~$10–$30
• Best method: 30-pin Bluetooth receiver

You’re basically reviving a dead hardware ecosystem with a $15 dongle, which is one of the rare cases where the modern solution is cleaner than any DIY workaround.