I have made a real-world test, not on the LME49710, since I don't have any sample in my drawers, but on the LM4562 which looks like an almost perfect clone, based on the characteristics.
The pathological behavior observed in sim was present, but to a lesser degree: the resonance peak was around 250kHz, against 400kHz in the sim, and the peak magnitude, both for gain and impedance was ~10dB, against 20dB in sim.
I made the test on a breadboard, and the 1µF capacitor was an ordinary 63V metallized mylar cap, which could explain the differences in part or totality: the breadboard adds contact resistances and parasitic inductances, and the cap is not the ideal one of the sim.
It could be that with a good PCB and a PP plain foil cap, reality and sim would converge -or not-.
Anyway, even if this expedient looks rather harmless, it is probably better to stick to more conventional solutions: monolithic voltage regulators are designed for the job, in particular capacitive loads, and they can be quite good regarding noise.
The LME is in principle better in this respect, but since it necessary to attenuate the reference voltage to reamplify it 10x afterwards, it mostly negates this superiority.
Initially, I made the test with ground-referenced inputs, but later I realized that using the opamp in a single-ended, regulator fashion could make a difference, and I modified the test setup accordingly, but no significant change was visible