I have a solutioon for your problem, but you will not like it. There are made LIFE batteries that are designed as direct replacements for the standard 7amp 12v SLA batteries that, on this sdide of the pond, are used commonly in alrm systems. They are in the same style case, with the same dimensions, but they weigh a fraction of the SLA, around 50+- % less. They have the advantage that they can provide high discharge currents, much higher than a standard SLA or even a high discharge SLA.
Te BAD news is the cost, on two counts. 1) the batteries are expensive, $180 amnd 2) they require a dedicated charger, around $30.
They do have an additional benefit, in that they do not self discharge as SLA's do, therefore their management is quite a bit easier. I use these in my 1/16th PT boat and they make keeping the weight in tow very nicely.
Sorry if this is a tease, but I am throwing the information out there.