Pulmonary Embolism Support Group
By far the most common form of pulmonary embolism is a thromboembolism, which occurs when a blood clot, generally a venous thrombus, becomes dislodged from its site of formation and embolizes to the arterial blood supply of one of the lungs. Symptoms may include difficulty breathing, pain during breathing, and more rarely circulatory instability and death.

If that's the case, I think your surgeon is probably on the right path. The clot has been there for four months at this point and is probably pretty stable. The chances of it breaking off are very slim. Clots are a lot like bricks held together with mortar. The longer they are there, the harder the mortar gets, which makes it pretty unlikely that your older clot poses much of a danger, especially if you are still on anticoagulants.
I will tell you that some DVTs hang in there long term. I had one in my right leg for about nine years. It was originally diagnosed in 2005. It was still there when a second DVT was diagnosed in 2009. (They told me at the time that it was basically scar tissue at that point and had bonded with the vein wall... no risk of breaking off and traveling.) About five years after that, I had another leg scan and the clot was gone. It finally disappeared somewhere between four and nine years after diagnosis, which was a surprise to me, since I thought I'd have it forever. It left a little scarring behind on the vein walls, but that's about it.
So, in summary, I think you're probably OK to have the filter removed even with a chronic clot in there. Your surgeon is on the lookout for NEW clotting, as he should be, which might change the decision to be made. But, if your filter is not made to be left in long-term, it's probably better to have it taken out now. The clot in your leg will probably eventually resolve, but it isn't worth the long-term dangers of the filter as protection against a stable clot that isn't likely to break off.
The actual removal took about 20 minutes. I was alert and watched the entire thing on the monitor. I felt some pressure in my abdomen when he captured it and was grateful for the meds. He put the filter in a specimen cup and let me take it home.
The filter is much larger than I expected. I will be on Eliquis for the rest of my life and if I ever need surgery in the future (hip or knee replacement) they will need to insert another filter at that time.
I am sore at the incision site but other than that I feel I will be back to work in a couple of days. I Thank God for a skilled surgeon and for the support from this board. Thank you Toss for your wise words.