Completely not.
Forget guarantees and warranties, they aren't really worth a jot. Under law as others have said, one you've accepted the goods - that is had the chance to receive and inspect it - any automatic 'right' to a refund is out the window. The only legal 'right' is a "right to remedy" but that does not automatically mean refund, and definitely doesn't mean you can simply claim a refund, rejecting all other offers of remedy out of hand. Consumer law is simply - and thankfully - not that lop-sided in favour of the customer. The trader simply has to offer a solution, nothing more.
This. I really really wish people would learn to differentiate between "fit for purpose" and "merchantable quality". If you ordered and paid for a battery (which dictionary.com describes as "A combination of two or more
cells electrically connected to work together to produce electric energy."), as long as it does that scientificy thing and matches any other mumbo jumbo of voltages and charge capacities, it is fit for purpose. End. Of. No ifs. No buts. No maybes. It doesn't matter whether it failed in 5 months, weeks, millenia or seconds. As long as you didn't also state that you also wanted the product to predict the World Cup winner or solve world hunger, it is fit for purpose.
What it arguably isn't is "of merchantable quality". I.e, it isn't built to a satisfactory standard to fulfil its purpose/function for a reasonably expected amount of time or usage. A dodgy ground to be on anyway with consumables, hence why they are very rarely guaranteed in the first place. But anyway.
Again, this. Unfortunately, you can't say simply say "I've already bought a new one, so just give my money back" It simply - in terms of 'legal right', doesn't work that way. The trader has to have a chance to offer a remedy.
Although, enough of erroneous presumptions and "can't dos".
Under the SOGA, any complaint of fault is split into two categories. After 6 months, the burden of proof lies with the customer to prove the fault has happened through technical fault rather than misuse or fair use. Within 6 months, the burden of proof lies with the trader to prove that the fault has occured through misuse of neglect. Otherwise,
the legal assumption is that the fault was inherent. That is to say, the fault was there from the start.
They've admitted already that there is a fault, so you could immediately question the P&P under responsibility of burden of proof. Also, they've admitted its faulty, but I'm assuming that they haven't provided evidence that the fault has occured since purchase. If they have or do do that, then the compensation has to be mitigated. If they can prove beyond a shadow of a doubt that the product worked initially, but something failed recently, they are entitled to say "well, you had 5 months use, so we'll give you x% back". Although I am not a legal professional, 60% seems rather Scrooge like. If they can't prove that, then under 6 months, the legal presumption is that the fault was there from day 1.
The bottom line is this; you can disagree with the remedies they've offered you. You may have further weight in that said remedies must not result in "significant inconvenience to the customer" (to use the legal wording). A battery for a vehicle used for business which the loss of would result in £x000s may - and I stress may - be found as significant inconvenience, but then the counter argument is could you not have used another vehicle? (for example).
You can disagree and then take it up with a small claim's court and argue "under 6 months", "not of merchantable quality" and perhaps "significant inconvenience from the trader-offered-remedies", and stand a fairly good chance of winning. But your awards in the event of a finding in your favour would be cost of battery + P&P charges + (at a push) costs. Potential business loses would not be taken into account as you have said you have already sorted an alternate means (so was it really a significant inconvenience?) and therefore don't have any quantified losses. And at the end of the day, is it worth the time and money?
In this instance, my own personal opinion, take the exchange.