I can’t modify the project right now because execution credits are unavailable, so no schematic components were added yet.
For your requirement — relay ON for ~1 second, OFF for ~10 seconds from a 3.3 V battery — a pure transistor/passive circuit is not reliable for repeat timing. The minimum practical circuit should use:
3.3 V battery
On/off switch in series with battery
Low-power CMOS 555 timer, e.g. TLC555/LMC555 type, configured as an astable timer
NPN transistor or logic-level NMOS to drive the relay coil
Flyback diode across relay coil
Timing resistors/capacitor for ~11 s period
Decoupling capacitor across timer supply
Important note: use a 3 V or 3.3 V relay coil. Many common 5 V relays will not switch reliably from a 3.3 V battery.
What remains to be done when credits are available:
Select a 3.3 V-compatible relay and CMOS 555 timer.