Some original records (School rolls and log-books) are held by the schools or their replacements, although few state schools existed during the dates you mention. There were private and church schools and you will be extremely lucky if you can find any records for these. Universal education for all did not occur until the last couple of decades of the nineteenth century. There are a couple of books on the history of Manx education that list schools by date and area - one of these may narrow your search a little.
If you can identify which school you may be interested in and can not find any records at the museum, you could try writing to the current headteacher for the school of the area (e.g. Albert Road School in Ramsey which was established in 1905) who should be able to tell you if any records exist - they are an overworked species but may be willing to help if they can.