If you have Illustrator, it would probably have been better to use that if you had known about this beforehand. You really should find a different printer, but here are some things you can do via Indesign and Illustrator.
In Indesign go to File/Export and pick EPS. In the next window you can leave the General stuff as is.. Click on Advanced and, under transparency flattener you can pick High Resolution or, if you’ve maintained your Illustrator logo as a vector, 100% vector. Save the eps.
Open the eps in Illustrator and make sure you yu get the thing centerd so it coesn’t get clipped . Under Effect menu go to Document Raster effects. Since you probably have a lot of text at a small size, you’ll want a very high resolution. While 600 dpi will get you through, 1200 will be better for fine text. Leave the background transparent and don’t check either of the options. Hit Okay.
Then export to Tiff. With the high resolution, the tiff should be HUGE but it should print ok.