As per datasheet, I have enabled LP oscillator. Why should I set the internal oscillator instead please?
if I understand correctly your intent is to use the intosc @4mhz for cpu clock and the ext osc @32khz for timer1 clk source

so to use the HFINTOSC @4mhz config= _INTRC_OSC_NOCLKOUT [no clk out assumed]
osccon= $60
Same question for T1CON.TMR1CS; why should I set this one?
to use the ext osc for timer1

you must enable the ext osc
bit 3 T1OSCEN: LP Oscillator Enable Control bit
and
select ext osc as timer clock source
bit 1 TMR1CS: Timer1 Clock Source Select bit

so t1con= %00111011 with 8:1 prescale and timer on