Unless my memory of the process is incorrect (others here may provide concrete example), my 2 cents are that once your application is "in process" (paper application, photos, documents, etc. received by proper authority), then any deadline on visa can be overrun, as the next step is just picking up your new ARC id with a new expiration date (which may start from the expiration day of your landing visa).
Here's link to phone hotlines: https://www.immigration.gov.tw/ct.asp?xItem=1092845&ctNode=32289&mp=2
You may either try one of those #s yourself on Monday (they have operators in English) or have your employer call on your behalf to get exact details.
Good luck. The light is at the end of the tunnel.