The LM317 has a drop out voltage of ~3V, so I doubt it regulates the battery output to the table. NiMH batteries need to be charged at 1C and negative delta V slope detection, thus the higher voltage input, the PIC processor and complex charging circuitry (another argument in favor of SLA batteries which are simpler to charge).
If the input of the table is designed for 12VDC, then 2 possible problems occur if you provide a higher voltage: Power dissipation and exceeding maximum voltage ratings on the input devices. Regulators (even low voltage output devices) typically have 18-26VDC input ratings. CMOS logic is usually rated to 15VDC (some CMOS devices are rated to 12V, but it would be bad practice to operate those at 12V so I doubt you find any devices with a max Vcc of 12VDC in a 12V table). Tantalum caps have standard voltage ratings of 6V, 10V, 16V, 25V or higher (6 & 10V devices couldn’t be used safely with 12V input so we can eliminate those). Power dissipation at 13.6 would only be ~13% higher than at 12V but 58% higher at 19V.
If the OP really wants to regulate the battery supply, then look at the LT3080 (or similar devices i.e. Micrel MIC29150 series which have fixed or adjustable outputs); the LT3080 is a LDO regulator (350mV) and 1.1A current rating. The output cap is critical for stable operation and should be a low ESR ceramic type located close to the output pin and ground.
If the input of the table is designed for 12VDC, then 2 possible problems occur if you provide a higher voltage: Power dissipation and exceeding maximum voltage ratings on the input devices. Regulators (even low voltage output devices) typically have 18-26VDC input ratings. CMOS logic is usually rated to 15VDC (some CMOS devices are rated to 12V, but it would be bad practice to operate those at 12V so I doubt you find any devices with a max Vcc of 12VDC in a 12V table). Tantalum caps have standard voltage ratings of 6V, 10V, 16V, 25V or higher (6 & 10V devices couldn’t be used safely with 12V input so we can eliminate those). Power dissipation at 13.6 would only be ~13% higher than at 12V but 58% higher at 19V.
If the OP really wants to regulate the battery supply, then look at the LT3080 (or similar devices i.e. Micrel MIC29150 series which have fixed or adjustable outputs); the LT3080 is a LDO regulator (350mV) and 1.1A current rating. The output cap is critical for stable operation and should be a low ESR ceramic type located close to the output pin and ground.