Bringing a 15+ year old 12 volt battery back from 1.5 volts is a huge ask, and i promise you the battery will never be reliable again.
If you attempt a bump start it is most definitely advised to do it on a downhill gradient in 2nd gear, otherwise it will be you that needs the jump leads.
Make sure all other electrics are switched off when doing so, like headlights, heated grips, spotlights, Xmas fairy lights, etc.
I will be bold though and say your battery has had it, and it's definitely time to treat the bike to an overdue Xmas present.
Don't go for the cheapest one you can find because it will end up as false economy. Buy a good branded name like Motobatt or Yuasa, and look for one that has extra cranking power, as they are tried and tested but still at a reasonably sensible price.
And be sure to look at the exact measurements of the battery in comparison to yours, as i've seen it before where they can somehow be totally different even though they're listed as suitable for your bike. I don't think the year of the bike will really matter, the battery compartments should all be the same on these bikes.
It says this is the right one, but i'd still do a quick measurement if i was you.
https://www.ebay.co.uk/itm/Yamaha-FZS-600-Fazer-2003-CT12B-4-CT12B-BS-Motobatt-Motorcycle-Battery-Upgrade/352456036581?epid=20025980731&hash=item52100450e5:g:N8wAAOSwqj9bkm7t:rk:10:pf:0