I don't see how the one with the BJT, in the original post, can possibly work. The current through the LED will be fixed at just over 9mA, at voltages over 11V and gradually fall, as the voltage goes about 11V, but won't turn off completely, until it's below the LED's forward voltage. The modified BJT one is much better.
I think falstad is being optimistic. According to the data sheet, the 555 timer uses around 8mA and would drain a 7.2Ah battery in just over 5 weeks.
https://www.ti.com/lit/ds/symlink/ne555.pdfIn the other thread, I recommended the TLV6703. If you want to use stripboard, then there are adaptors, but it's possible to use SOT23 on stripboard, although it's not the easiest thing to do. Cut two tracks, bend pins 2 & 5, on the device upwards, solder pins 1, 3, 4 & 5 to the board and connect pins 2 & 5 with 32 AWG wire and make it as short, as possible.
I'd just use the adaptor though.
https://www.sparkfun.com/products/717https://www.eevblog.com/forum/beginners/the-simplest-12v-low-battery-chirp-possible-which-off-the-shelf-components/msg3401354/#msg3401354EDIT:
Do you want the LED to turn on, or off, as the battery falls? The TLV6703 will turn it on, when the battery voltage is low. The other circuits posted here will turn it off. I you want it to turn off, then use the TLV6713.
Here's a demonstration of how to use the TLV6703/6713. For a threshold voltage of 10.5V use 10M for R1 and 390k for R2. R3 limits the current through the LED. Modern LEDs are fairly bright, so might find 10k is adequate.