This is a brief tip, untested but potentially saving you bucks.

If you want to pay around $165-170 you can get a replacement solenoid direct from Suzuki or a dealer.
At a wreckers they might try and hit you for upwards of $75 for a second hand one...

In Australia or NZ, the cheapest solution might be to obtain a solenoid from Jaycar electronics.

Their SS-0903 solenoid (AUD $29.50 feb 2004) is a 12 volt tube type push solenoid and it's roughly the
same dimensions as that found in the across, so could fit into the brackets that are on the bike.
Some minor auto-electrical work would be necessary to mate up the contacts.
I haven't tried this, but I have seen the solenoid and I know that it looks like it will do the job.

The push type exist on the rear, fuel door solenoid. The boot ("handbag holder" for the girls) solenoid
is a PULL action. At the time of writing Jaycar has a square one which is stronger and cheaper (22.50) than the above unit, and this might do the job.

If you do the mod with this part, please email me so that I can let others know of its success / failure.

Thanks to Emma for the incentive to update this page.