I must fit a coolant reservoir to my 8-year old Beta 13hp.

Beta engines are used around the world, often in much warmer waters than we experience in the UK. I can't imagine the heat exchanger is under-specified, nor can I believe that the small additional amount of coolant allowed by having an expansion tank would make any significant difference to the cooling efficiency.
 
Getting back to Old Rhodie's problem, if there is plenty of seawater (coming from the exhaust I assume) and the heat exchanger is clean, then there must be either a blockage somewhere in the fresh water circulation OR the heat exchanger is undersized OR the water on the freshwater side is being lost by expansion and uncovering some of the tubes in the HEx. A header tank will solve only the last problem but is the easiest and cheapest fix to implement. Searching for "radiator expansion tank" threw up lots of options of which this one http://www.ebay.co.uk/itm/RENAULT-C...39&rk=3&rkt=4&sd=150898784461&#ht_1668wt_1091 looks the cheapest and smallest. But try to have a look at a Clio first to see what size it really is.

As has been said already, talking to Beta would be an idea.

I fitted a Corsa one and it looks lower than the Clio's.
 
In the installation instructions, Beta specify an inlet seacock of 1" BSP, presumably any less than this may result in insufficient flow at high rpm. When my Beta 20 was installed the seacock and inlet strainer were replaced to the spec advised. I have never had any cooling issues even when motoring hard for fairly long periods.
 
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