I bought the EM1.3 solely for my 4 FT lenses and to get a longer reach too.
There are other nice features on the EM1.3 that we birder may not use but I found the Pro Capture High feature very useful. Infect this feature compliments the A9 very well. With 20fps we get many action shots but with 60 fps we get 3 times more action shots compare to the A9. I recently shot Barbet nesting and man the different action shots I got I could not get close with the A9.
Here are samples shot with EM1.3/300f2.8 no TC
https://www.flickr.com/photos/ricseet/49749171908/in/dateposted-family/
https://www.flickr.com/photos/ricseet/49647806641/in/dateposted-family/
Yes it works very well with the 1.4 and 2.0 TCs. I often shot with the 2.0 TC.
Here are 2 samples
https://www.flickr.com/photos/ricseet/49620758678/in/dateposted-family/
https://www.flickr.com/photos/ricseet/49621455371/in/dateposted-family/
These were shot with the EM1.3/300f2.8. I recall you have the 7R3/100-400 and you may wish to pitch it against the EM1.3/300f2.8.
When the 2X TC is still insufficient reach for smaller birds I often resort to 2X digital Zoom. Here is one taken with EM1.2/300f2.8 with 2X TC and 2X Digital Zoom. Original image was soft as I was not able to do a good job calibrating the lens plus the extended reach. For a record shot it decent.
https://www.flickr.com/photos/ricseet/49231596838/in/dateposted-family/
No brainer get the EM1.3 and your Oly system will compliment your Sony very well. For starter extended reach and awesome IQ. Then there is Pro Capture which gives you lots of joy doing super fast action BIF shots.. I have tested the system for tracking BIF and I have better keepers compared to EM1.2. But A9 is still King Of The Sky. With more practise I am sure it can be improved. Danny from m4/3 DPR Forum takes great BIF using his EM1.2 with Canon 600f4?. Gear is important and Danny has the skills for BIF which I am unlikely to acquire in this life time. 76 this coming August. Thats why I bought into Sony to close my skill gap for BIF. LOL.
BTW Can you please advise how I can link my Flickr postings to this forum. I will PP the Barbet shots to share with you.
Thank you
Stay safe everyone.
ric