I had to reply to your topic, I can totally relate having been there myself, it's not an ideal situation but adjustments can be made. I still stand by my method, but the decision is yours to which way you go and I wish you the best.
In relation to your cycling/ water change question I would avoid doing 100% water changes, I believe they are just too stressful. (I personally hate this option)
Using an already establish filter would be great if you have one, if not I would at least try to get the tank to cycle, it should be possible if you keep with it.
Just keep an eye on the ammonia and nitrite levels (I assume you have the liquid test kits, if not I highly recommend you get them). For every .25 of ammonia in the tank do a 25% water change or for every .5 level of ammonia do a 50% water change. Just make sure you have Prime and dose every 24hrs the correct amount and your fish will not be effected by those levels of ammonia. It is safe to dose every 24hrs using prime providing you use the correct dose. I have done this with all my fish. They were all "fish in cycle" which is safe for betta's providing done properly.
My guy used to love his big tank before he got SBD, there was no stopping him, but now he appreciates the low water level, I have experienced this first hand and I strongly recommend you keep the water level really low - like 10cm low, others may disagree with this but from my experience it was the best thing I did for him (with his floats of course) however there is Swim Bladder Disease (infection) along with Swim Bladder Disorder and then of course varying degrees of severity. My guy, after 3 months, I can safely say it is a disorder, still inquisitive and eats well, still fiesty so he could go on for a very long time yet but every case is different. If it is more serious it will show itself sooner rather then later.
Epsom salts will help if the cause of his SBD is constipation I tried fasting my guys for 3 days and did the epsom salt treatment however this did not help with his SBD. So there was no point continuing. I have noticed if I feed him more he sinks quicker so I do feed him a little less then what I used to for this reason.
The only difference is is that I kept him in his original tank and dramatically lowered the water level. If he is going in a smaller/different tank it may take him a little longer to adjust, but I think long term, maybe even short term he will appreciate it.
Being pessimistic here but if he get's worse ie: stops eating there may be something else going wrong internally with him and you may need to consider putting him down (if he is suffering and incurable) But, that's absolute worse case scenario. On an unrelated topic I had to do this once with a betta that had dropsy and after 5 days I just couldn't bear it anymore and did the clove oil thing) I hope I never have a case of Dropsy again it is the most hideous disease and always ends in fatality but I wanted to give the little guy a chance but it got to the point where there was no choice, the pinconing and swollen gut looks and I imagine would be extremely painful.