Looks like you have a wonderful new friend! I don't see any compromise there.
I might be able to lash the cedar fence to the chainlink fence with rope and solve the problem. Rope might loosen and is chewable.
I don't know. Advice?
Advice from a person that owns and has owned several Siberian Huskies, who are considered to be escape artists extraordinaire. Also a massive Malamute that could bull through stuff. We only let them cruise the back yard when we were home, awake, and could monitor them.
Use wire fencing to fill the gap. Your buddy will simply floss his teeth on rope and cable ties. Get a spool of 4 or 5 ft high welded fencing used for gardens. Double it over about 2 ft on a side, bend it into a long L. Fence-staple it to the wood every 4 to 6 inches. Wire it to the chain link every 4 to 6 inches with bits of wire you clip from the welded fence scraps. If your original fence is higher than the welded fence, create another doubled - over L section and lap it over until it reaches the top.
If your dog starts to dig underneath, add dig guards. Use the same 4 ft welded wire fencing. Cut it into convenient 6 or 8 ft lengths so you can manage it. Bend it into a loooong L, 2 ft on a side. Lay it on the ground and fasten the vertical portion to wood or wire fence every foot or so. The wire on the ground is very annoying to dogs' paws when they try to dig through it, and usually they stop. Usually
Extreme measures... electric livestock fence
Our Sibbies jumped over, dug under, and then simply pulled apart the welded wire fence. We added 2 strands of electric livestock wire, 6 inches and 18 inches high. The physical fence slowed them down. The electric fence popped them every 1 second and was totally annoying when they were engaged trying to bust through the welded wire. Yes, 3000 volts hurts. I tried it on myself first before my dogs. It stopped them until one of them pried loose a dig guard, folded it down, shorted out the electric fence, and dug under the shed to get the skunks.
BTW, the first AKC recognized Alaskan Malamute was named "Rowdy" because he was very, very calm.