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It may be that the battery in the Garmin is dead, if it really does not respond to the charger in any way, nor show any signs of life.
Is that how it is behaving? or are there indicator lights, or beeps, that show up on it when it is turned On?
That is regrettable. It means that the Garmin S1 will either have to be service-repaired, or be replaced, perhaps. But there is still one other thing you could try:
Would you like to attempt replacing the battery in the S1 yourself
OK. Would you be interested in Garmin USA's fixed-price for repair (of any kind, even just battery replacement) for your Approach S1 device?
The Garmin quoted 'flat-rate' price for the repair of an Approach S1 is