ZBurnett,
As you have experienced, the Lombard will boot nicely to an optical disk without the HD. How long does the Lombard run from the CD and then what happens...freeze?...no display?...etc?
You certainly have a hardware problem if it will not stay running when booted to a CD. I might suggest this approach to narrowing down the possible culprits:
1. Remove the main battery and optical drive; now connect just the power adapter and power up. Do you get the expected flashing '?' on a gray screen?
2. Next, lift off the keyboard, then carefully disconnect the keyboard ribbon cable; a bad keyboard can cause no-starts, freezes, etc. Test the Lombard by inserting the optical drive with bootable CD, then power on. Technically the powerbook should pick up the CD as a bootable volume and boot to it, but I can't be certain.
3. I would next disconnect the PRAM battery and leave it disconnected for the duration of testing; the 'book will run fine without it. If the rechargeable PRAM battery is failing internally, it can play havoc with the power manager. Now test your 'book.
4. Remove the top memory module, then test.
5. Lift out the microprocessor card, turn it over and remove the bottom memory module; now reinstall the microprocessor card (bottom slot empty) and reinstall the original top memory module (if it made no difference when removed earlier). Now test...
If you are trying to boot a Jag/10.2 CD, it may boot with just a 64MB module but I don't think Panther/10.3 will. I mention this since I don't know what your RAM specs are. If you have any MacOS 8.6 > 9.2.2 compatible CDs, it may be a little easier to troubleshoot.
If your main battery is dead, don't rule out the possibility of an intermittently failing power adapter. The PB1400, 3400, any black G3, and first generation iBook adapters are interchangeable, so it might be worth a try.
Here is a nice disassembly guide:
http://www.ifixit.com/Guide/Mac/PowerBook-G3-Lombard/31/