In order to make that critical high-value transaction you are going to have to move a high volume of BTC to the sidechain
Unless you buy BTC on the sidechain to start with. For example, they spend BTC to buy mining hardware to mine anonymously. Isn't that how you got your XMR.
It's hard to envision how this works in the foreseeable future. If you buy mining equipment and mine the merged-mined sidechain, the vast majority of your rewards will still be main chain BTC. You will get only some small fees in sidechain coins.