However, being small and cheap, that device is pretty noisy, particularly with regard to common mode noise.
... it just needs to be simple, easy and small... How hard can that be?
There's the rub. Simple/easy - small - cheap? - low noise - efficient. You'll have to compromise between these somewhat. As far as I know you can't avoid using "something switching" to get +/-15V from 5V. However, there are varying degree's of "noisy" both in terms of conducted and emitted EMI.
Royer converters
can be quite low noise (but often aren't designed to be) with reasonable filtering, and only require a handful of components, but finding an off-the-shelf-transformer for the job is hard, and often requires custom magnetics. Simple, small, quiet, not easy.
A boost converter with either a tapped secondary or a dual charge pump can be reasonably small, efficient, simple to work with, but you'll probably want higher frequency for easier filtering, and careful layout. A tapped secondary isn't easy to find, but easier than the above royer because the converters regulation allows for a range of turns ratios. a 1:1:1 to 1:4:4 is fine. Not so simple, relatively easy, not so quiet, small.
Charge pumps themselves tend to have lower EMI (no ferrites), and higher efficiency. But for a 3x boost @1W I haven't seen a single chip solution capable of that, which is why I suggested using them with a boost converter, either to create the negative rail, or for true symmetry, both rails running off the switch node. of a 5->10V converter.
In the past I've tended towards a small boost converters + charge pumps, but that was for mostly opamp circuits where the CMRR came in handy.