What power source and amplifier should I use for high reactive power purpose?
If you have an efficient 4 quadrant (and it's the 4 quadrant that's the important bit) switch mode converter for your 'amplifier', then in principle, your ultimate wall power supply will indeed be the 20w coil losses plus the residual converter losses. What the converter will do is recharge the its input capacitors when the load is delivering power back to the supply.

If you have a conventional linear amplifier, that uses dissipation in beefy transistors to power the loudspeaker, then no, it has to be supplied with the full load VA, and then some for losses. 

A 1kW amplifier will handle it, right?

No. As the voltage across its series pass transistors will be much more when handling the reverse power than its designers anticipated for conventional loudspeaker use, it will get much hotter than expected. All that 1kW input is getting lost in the amplifier, practically nothing lost in the load. You may get away with extreme forced cooling, as it's only the power dissipation that is high, the peak volts and current are still OK. You would need an amplifier rated at 3kW output power into a conventional resistive load to handle the 1kW of internal dissipation safely without modification. A good class AB linear amplifier rated at 1kW output will only be dissipating around 300W internally.

For a further fly in that ointment, although a 3kW rated amplifier will have the dissipation, it will use a higher internal voltage than you need (to deliver 3 rather than 1kwatt) so will still dissipate excessively. To keep the excess voltage down, look for an amplifier rated for a lower impedance than you apparently need. As a hifi buyer, I'd look at amps rated into 4 or even 2\$\Omega\$ and comment something about specmanship and my speakers are 8\$\Omega\$, but in your case, high power at lower impedance will work to your advantage.

We haven't addressed the drive impedance of your load, what are the peak current and peak volts required? This may well affect things like should you use a transformer, or multiple amplifiers in parallel.

If you use a full H bridge motor controller, it will almost certainly have 4 quadrant control, it's designed for putting braking power back into the batteries. I doubt that the stock bandwidth will stretch to 100Hz however, but it might, and if it doesn't, it might be hackable with a few timing component changes.

A class D switching audio amplifier will have the bandwidth, but will it have 2 or 4 quadrant control? With 2 quadrant control, it may interpret the reverse power from the load as a 'pathological load', and shut down. Different manufacturers may use different control algorithms for their chips, some may use 4 quadrant, it would be worth researching several. 1kW is a lot of power to save! Unfortunately, the standard way that audio amplifiers deal with 'difficult' loads is to add extra resistors, for instance a Zobel network, that make the load 'more' resistive, which is exactly what you don't want.

You may be able to speed hack a motor controller, or you may have to brew your own converter from scratch. At 1kW throughput, it's not a job for the faint hearted, though the low 100Hz bandwidth does mean that it should be easy to repurpose components designed for class D audio use.

It may be worth tuning your load for 30Hz, it will still help rather than hinder at the frequency extremes, just not very much. Can you switch capacitors with relays or power FETs for better compensation over smaller ranges?