I'd use some dollar store 9V batteries to power the OPAMP or a separate floating PSU and a rail splitter, as your diodes make the already excessive headroom required by the LM317 positively ridiculous.
Lets suppose you want to push it up to the LM317's design load current of 1.5A - worst case, you need 2.5V headroom for the regulator, you have another 0.15V drop across the 0.1R resistor + >2.8V across your diode stack, and you are up to a 5.5V minimum input voltage.
If you use a floating supply for the OPAMP, the minimum input voltage will be under 3V and the minimum current will be down around 20mA, maybe lower.
If possible use a floating 12V supply so you can easily fan-cool the LM317's heatsink - it will need it. For the rail splitter, if you don't have anything better, use another OPAMP as a unit gain buffer, with its output to Gnd and 2x (10K || 0.1uF) as a potential divider across the 12V supply and its output to Gnd. As OPAMPs dont tolerate capacitive loads, it is *CRITICALLY* important that you don't put any extra decoupling between the terminals of the floating 12V supply and Gnd. You can (and should) however put as much as you like directly across the resulting +6V and -6V OPAMP supply rails, especially if the floating supply is also running a fan.
LTspice sim attached.