After enjoying the RIAA phono preamp for several weeks with the new battery pack, I decided to build the battery charger just not to be surprised by the lead-acid battery pack running out unexpectedly.
The lead-acid battery pack is formed by a 12V and a 6V 1.3Ah batteries. Between both batteries there is a total of 6 +3 cells = 9 cells. Each cell requires 2.27-2.30V of charge at 15-24C. Therefore the charging voltage should be 20.43-20.7V. Exceeding this voltage will reduce the battery life. The charging circuit has to be constant voltage, not current.
Rod recommended me a simple but very effective design based on the TL431a. The regulator output voltage is set by R1 and R3. 0.1% precision resistors are needed, otherwise a trimpot should be used as R1. R3 should provide at least 2mA for both Q1 and TL431a operation: 
A simple PCB was built in less than 1 hour. You need to place the BD439 in a heatsink as it will get hot when charging the battery:
In a question of 2 hours the battery was charged back fully. I used it for around 20 days so far I think. The initial current is as high as 370mA and drops as the battery charges and the voltage goes up. The residual charged voltage was about 19.4V. This drops quickly after some current starts to flow through the preamp.
