Hi Olaf,
Please look at H2 docs "Performance"
page. It shows that 10-15 times slowdown is kind of expected, when you switch from local to a remote server mode.
On the hand, 1000 times slowdown does not look right. It might be you network performance (unexpected routing via proxy etc.) or it could be H2 misconfiguration, because. 3000 queries/sec does not sound like much, assuming your queries are simple few rows selects with proper indexing.
As far as memory goes, you can try to increase heap size as much as you can just to see if it makes a difference.
In any case it is impossible to give any meaningful advice without knowing more details.
You may also try to run some provided benchmark, mentioned on that page to see whether your results will be in the same ballpark.
Andrei