First, my official line:
If the center of the root ball is staying too dry, the repot is NOT likely to be "gentle". You're likely going to need to be breaking into that center ball and breaking it up to get fresh soil into there. That's a lot of root damage. Any amount of fresh soil around the dry root ball doesn't really help much
So, given that, one half-way measure that could get you to a safer repot time might be to get a solid chopstick/dowel/metal rod/thin spike/etc...and bore some drain holes into the dry hard center to allow air/water to get into there easier. Think what the landscape guy does to the lawn for the grass. You might be able to open up the soil quite a bit that way without having to repot now. Unless the tree appears in distress, you could even start with drilling in a few holes wait for a week or two and see how it goes before drilling in a few more or not. I have done this for some of my trees to get them through a season or two until a safer time to repot.
Now, my un-official line which probably aligns more with what you
WANT to hear even though it's widely regarded "a bad idea".
I do not endorse this approach.
However...winter before last my Key lime was loosing a lot of leaves without new growth going into the winter months. It was pretty spindly. I was worried about the soil I had it in and if the roots were suffering...all the same anxious thoughts you're having, etc... So, I repotted it in late December...Christmas day in fact because my wife was working that day and we didn't celebrate Christmas until a week later. That tree was in a big 5 gallon porch pot...nothing a respectable bonsai would ever get anywhere near. One of those vase shaped can't slide the dirt straight out must much around with a spade and pruning shears kind of pots. I mention all that because I can assure you...
I disturbed the roots getting this plant out of there! The tree did live. I got over 100 limes off it this summer!
BUT...what I learned was 1) the soil was fine...easily good enough to make it through the winter though from just what I had seen from the surface, I had my doubts. 2) The roots were fine...not a problem there at all. 3) If I worried about my own children this much they'd all be walking around in bubbles with tip sensors that alerted my phone whenever they laid down! 4) Now I was not only worrying about the
initial problem (lack of growth and losing old leaves) but
NOW I had the additional worry that my ill advised repot killed it for sure! 5) Sometimes rules are made to be broken.
There are a lot of potential lessons one could take from my un-official line. I'll let you pick the one most suited to your frame of mind. I would encourage, however, that you consider less invasive practices such as my official line or whatever the good folks on this site recommend
first. A tree depotted will not pot itself