To get perfect TC you'll have to "black & code" the beginning of the Beta tape using the deck. What this means is recording black and using the Preset Timecode feature on the deck to set the starting TC. For example, most tapes for b'cast start at 00:58:00;00 and have 30sec of black, 60sec of bars & tone, 10sec of black, 10sec of slate, 10sec of black, and the show starts exactly at 01:00:00;00. To do this you'd do the following things:
Step 1: Set your FCP timeline to start at 00:58:00;00 and make sure the black slugs, bars & tone, slate, and show all start at the proper timecodes.
Step 2: Black & code your tape. Preset the TC on the deck to start at 00:58:00;00, make sure only black is coming into the deck, record 30 seconds or so then rewind the tape back to the beginning and change the TC on the deck to "regen". If you don't know how to use the TC features on the deck consult the manual.
Step 3: In FCP put the In Point in your Timeline to 00:58:20;00 or so (you need to give the tape some pre-roll space), properly route FCP to the deck and Use the Edit to Tape feature to lay down your show. You'll also want to leave 10 or 20 secs of black at the end of your timeline. If you don't know how to use the Edit to Tape feature in FCP consult the manual.
For compression you'd ideally want to be working in an Uncompressed SD sequence. The proper Easy Setup is probably something like Blackmagic Uncompressed NTSC or something like that.
FCP might not be frame accurate so to correct that you can adjust the Timecode Offset feature in FCP.
Final Cut Pro: How to Calibrate Serial Timecode
Lethal