sparse column or DATA_COMPRESSION?
In the MS documentation, there is a chart to estimate space savings for sparse columns.

According to said chart, if at least 86% of the values are NULL, you will have an overall savings of at least 40% for this column. Each TINYINT value is going to take up 5 bytes versus the normal 1 byte in this scenario. 

You can compare this with the compression savings by using sp_estimate_data_compression_savings for the proposed index.

This should give you enough information to make an informed decision about which is the right approach.

Resources:


sp_sizeoptimiser Uses statistics to automatically recommend changing a column to sparse if it meets the threshold to reclaim at least 40% space. Disclaimer: I wrote it!