We received and answered this question via email. For everyone else's benefit, I'm pasting the reply we sent via email below. The table that stores custom asset fields is tblAssetCustom.
The only way to increase the number of custom asset fields would be to add fields directly to the underlying Lansweeper database, which is unfortunately not something we can provide instructions or support for. Custom database changes can break your installation and cause issues during updates. We've added your request for additional asset fields to our customer wish list, but do not have an estimated release date for it at this time.