There was no need for the capital ships to bombard the coast line , because the British didn't intend to defend it. They had coastal gun batteries and strong points set up, but these never stopped the germans before and wouldn't have done there either. The First line German infantry divisions landed were reinforced to the point of looking like semi motorised infantry divisions.
British General Allan Brooks correctly recognised that his main chance was to wait until the Nazi land and focus on that spot for a counter attack. Since only 1/2 of his divisions were even fit for combat he had to be careful to gather up his forces. The RAF was going to have to bomb the beaches in the hopes of stemming the tide.The plan was for each landing barge to have a gun platform mounted overhead where army infantry guns, flak and PAK were to be carried and provide what ever fire support they needed and then be 'dismounted' and moved inland after the beaches were secured.
The Stuka would have made short work of the Coastal batteries, while the BoB rages over southern England. At the same time other Stuka wings would be progressively reducing the RN anti invasion fleets , even in their ports. Given the record on these attacks in the summer fall of 1940 , it would probably take 2-3 weeks to neutralise the RN flottilas in the south coast region. While that was going on , the Sealion plan envisaged the KM setting up extremely dense mine barriers [1 mine ever 7-9m] , that should slow the RN considerably and cost them many many ships to cross each day, since the Germans had enough reserve mines to re mine the breaches every day for months if needed.
Plan for use of German captial ships was to sortie ahead of the invasion into the North Atlantic as in Norway, to draw off the RN homefleet. Given that they have months of endurance with replenishment at sea, they could surge into the North Atlantic weeks ahead of the planned invasion and lead the 'Home Fleet' on a wild goose chase. It would be better if they went of in smaller groups so the more capital ships the bigger problem for the Admiralty. Churchill and the Admiralty were terrified of the prospect of these surface raiders scattering a convoy, to be left to be picked of by Uboat wolf packs. At any given time their may have been a dozen convoys on route to the UK, so the Nazi could have had a field day.
Remember the overreactions when Graff Spee sortied into the South Atlantic and the Twins sortied later that winter. The admiralty panic and sortied the entire home fleet.Mean while Admiral Scheer slipped right through the screens right under their noses back into Germany. Admiralty misread Norway and the Germans got away with an invasion of the country. Raeder had done his home work on the RN , knowing that they were bound to their ports and had to react to German surface raiders, under political pressure. The Kreigsmarine had realised that two divergant naval squadrons/groups could act similtaneously with counter moves seas apart [even oceans apart] forcing the Admiralty to commit valuable reserves on wild goose chases. This would allow them to gain temporary control of a given sea long enough to cross it with troops.
So yes the bigger the capital ship fleet the more obsessed the Admiralty would have been. They may have caved to Admiral Forbes demand to cut back the anti invasion destroyers by 1/3 to provide more escorts for convoy duties. They certainly didn't want to have to mix it up over the channel since they might suffer crippled capital ships at the hands of the Luftwaffe. As far as the Admiralty was concern the anti invasion task was the job of the RAF and the army, not the RN. They were already stretched to the max with their empire and convoy committments.