Usually, combining (L)GPL software with other software into a single program makes that program subject to the (L)GPL. However, the copyright holder to the (L)GPL part can issue an exception to the (L)GPL that allows their software to be linked with software that would otherwise not be (L)GPL-compatible. The resulting program is not subject to the (L)GPL.
Here, the open source project uses the LGPL 2.1 license. You may use that that code freely for commercial or non-commercial projects, in accordance with the license.
However, once you also link to the Epson binaries you have to accept their license, which seems to be non-free. Notably, it does not allow sublicensing.
Since it is not a free license a discussion of its terms and conditions would not be on-topic here.
But it seems clear that you cannot publish a software that includes the Epson blobs. Your options are:
using the open-source code in a way that does not rely on these blobs, in which case you have no restrictions beyond those specified in the LGPL.
publishing a software that may depend on the Epson blobs, but is not yet linked to them. How you may do that depends on the Epson license. For example, you may be allowed to create a tool that asks the end user to accept the Epson license, then downloads and links the blobs for them. Again, whether this is allowed at all and specifically for commercial software depends entirely on the Epson license.
Note that the open-source library itself seems to fall under this case since it is published under the LGPL which allows commercial use and distribution with commercial software. The library already includes tooling to link with the Epson blobs if they are installed on the system (which requires their license to have been accepted). However, it does not assist with installation of the Epson blobs. So at the very least you seem to be able to offer a commercial software including the LGPL component if it links to the Epson blobs on the end user's system, not before.
