The segment data is sent as the 7 segment data for all displays including alpha-numeric, and the extended segment data is sent as well on different lines on the same strobe (for the 2 alpha numeric 14 segment displays in the 4x7, or all if it's the 2x16). As dumbass noted, the 4x7 displays' data are inverted vs. Data East and the 2x16 displays, but you can flip that easily enough when you grab the data. IIRC all the strobes are present in BOTH systems, so even with a 4x7, it's still strobing 16 times. (You'd only need 14 if you were just sending 4x7 as 2 digit pairs per strobe).
The circuit that's going to grab the data would be something like this:
Connect up all the segment lines to inputs. (14 segments per character, +1 bit for the period and +1 bit for the comma extension, IIRC)
Connect up all the strobe lines to inputs.
Watch the strobe lines and when one strobes, grab the segment data. Depending on which one strobes, you know which digit it is. You need to store 32 bits of segment data per character position, top and bottom. Invert it if it's the earlier 4x7 displays.
Outputting to your LCD is something I know zip about, but you're going to just take the segment data and combine it into characters and insert it into the appropriate position on the display. Also, I don't remember which special strobe is used to handle 'extra' displays like the credit display on the very early games, or the extra display on Taxi for instance. There might be an extra set of segment data for those machines and they merely do 3 sets of segment data on certain strobes.
The strobe ends up being your trigger at any rate - if you grab the segment data after the strobe, you should be able to write a glue logic program to output to the LCD (especially if it's an LCD that lets you do position manipulation, i.e. you can tell it, put this segment data at this position and don't change anything else).
If you're short on input pins on your platform you could possibly come up with some logic to take the 16 strobes and crunch them down to 4 bits and just read/react to that.
Now, it looks like the OP wanted to modify the actual DE code instead to run the LCD panel directly - that could be done as well but you better know 6800 assembly language and be willing to disassemble quite a bit of the original code to change that one. As a bonus in that case you can change the way the segment data is output. I don't think you're going to find something plug and play though.... there's going to be either glue logic in terms of decoder chips, or software; or modification of the actual data east code. pinballcode.com has done a lot of extensive work on data east games and would be most familiar with how it's internally constructed, maybe you can get some hints from him.
Ed Cheung's williams tester from a long time ago used a small 2x20 display to show the displays, and that was way back when, so definitely doable that way, although that was probably a system7 or 9 tester - so no extended segment data to worry about there.