Yep, correct. Too much of a swing in PH and Temperature, as well as other water borne minerals, can literally shock the fish.
Water changes are also really stressful and you should never remove the fish when you do it. Frequent hoiking in and out of the water will cause him no end of hassle and discomfort, and stress is one of the biggest causes of disease.
Whitespot is often triggered by stress. Uusally when in combination with poor water quality.
Small frequent changes will keeps things at a constant.
What you could do is 25% on a Monday, 25 on a Wednesday, and 30-50 on a saturday as a little extra.
25% on a tank as tiny as that is plenty for each change - I have one that size with live plants and shrimp in it, as shrimp is all a tank so small can support .
My main tank is 30 gallons and has a community and live plants in it. And yes, there is a betta in there
. The mix of fish I have with mine is very unusual for betta tankmates) and not normally reccomended for a beginner. I know my fish really well and have several spare tanks if something did go wrong.
Once you've upgraded to a larger tank,
keep the smaller one. You can use it as a spare in case something happens to the larger one ( a crack for example ) , or set it up with invertebrates like shrimp if you want to keep it running, which I have with mine. I also have spare filters and heaters in case of failure but I'm slightly pedantic like that.
While a 5 gallon tank can only support one betta and no other fish, if you
are able to go larger than that, 10 gallons for example, there is a small selection of safe tankmates you could try keeping with him, depending on his personality. Ricefish, micro rasboros, normans lampeyes and dwarf cories ( not all at once ) are some good peaceful tankmates, and not too colourful, so they shouldn't be attacked by him.