I think we've identified the main issue, Landlubber: You aren't willing to get 'stuck-in' with your capital ships, which prevents them from bringing their firepower to bear (Shrikes and Battleship). I'm guessing you have tried to get 'stuck-in', but it didn't work out, so now you're not willing to commit properly. Perhaps it was luck, perhaps it was tactics (yes, there are squad tactics in FSA), maybe both.
I'm also guessing the Shrike playstyle doesn't work for you, but maybe the Heavy Cruisers playstyle will. These guys like to stay away and really pound their opponents before getting up close for the kill. Basically, they should be putting out more firepower than their opponent at all ranges, particularly at long range, which might appeal to you more. If they can get a RB4 shot on Turn 1, and slowly close the distance while firing every turn, they'll be happy.
As for the Battleship, DR 6/CR10 with 2 Shields is Standard, so DR7/CR11 with 1 Shield is generally superior. Essentially, one of the Battleship's shield dice ALWAYS rolls a success. I'll take these defenses any day. The Battleship has basically the exact same tactics as the Heavy Cruisers too. Pounds from a distance, then kill up close. The difference is the Battleship has much better defenses, Wing Capacity, Mines, extra Fore Beams at 0"-20", and a couple of options, for slightly less firepower. However, lets look at the real cost:
3x Heavy Cruisers: 270 points base, max AD at 24" = 18
Battleship + Heavy Cruiser: 280 points base, max AD at 24" = 16
16 AD is when you start having a reasonable chance to Crit a Battleship, but 18 is reliable. That said, the Heavy Cruisers are generally better off firing three shots of 9AD at Cruisers, and potentially landing three Critical Hits. I'll take 6 HP on Cruisers over 2 HP on a Tier 1 any day.
The real Battleship cost comes from the upgrades and hard points. You should always take 4 Wings and Special Forces for +30 points (don't forget the Heavy Cruiser). Add Hardpoints as your points allow; my favorites are +2 AP and -1" Turn Limit for another +15 points, for a total of 325. This is a lot for a Tier 1, but you have to accept you're playing an elite fleet with above-average stats across the board.
So lets look at a list:
310: Battleship + HC
270: 3x HC
100: 4x FF
100: 4x FF
That's 780 points, leaving 20 points to upgrade the Battleship or Heavy Cruisers. I'm inclined to dump them on the Battleship, but Spook Torps for the Heavy Cruisers is good too. I'd also lean towards Interceptors over Fighters or Bombers, because I really enjoy shutting down all Torps my opponent might lob my way.
Fleet Tactics: I'd generally place a squad of Frigates in Reserve, because they're a great throw-away unit with good damage potential when they arrive... and their early absence isn't very detrimental. However, don't do this if it will lead to your opponent having 2 more activations than you Turn 1. Another option is to place one squadron of frigates in one corner, and another squadron of frigates in another corner. Assuming you won the first FTB roll-off, this will force your opponent to place three units before you commit your forces to a corner. The goal is to make the game as much about a full corner-to-corner game as possible.
With Interceptors, you should be able to easily guard nearby ships with their +8 to PD bubble; a 12" diameter PD bubble. Make minimum moves towards your opponent, and be patient. Your Direct Weapons have more AD than theirs, and you should be threatening Torp damage while you're blocking everything. Sacrifice the Frigates to force your opponent to activate, so when you activate, a minimum move will allow you to barely cross the RB barrier. Don't make a full move to cross this barrier; instead, just hit that RB next turn. Generally, move the Heavy Cruisers last, so distance can protect them from damage, and move the Interceptor Token so it can guard them before and after their move.
Turn 3 is when you will probably hit RB 2 with your entire fleet, ideally with little damage, but some noticeable damage on your opponent. You could potentially inflict 10+ HP of damage on Capital ships this Turn, which is significant at this point level. You'll probably loose a Heavy Cruiser and half of your Frigates, and probably have some Battleship damage, but ideally, your opponent will have even less to work with. You might even have the opportunity to Board something with your Battleship + HC, which could be 11AD of Special Forces. That's an average of 11 successes, which will threaten anything with a Crit, particularly if your Kinetics landed some Crits and caused CP loss. Unless something crazy happens, you're probably not going to capture anything, but sometimes those fanatic RSN Space Marines really pull through. (What do you mean no one told you the RSN SF are 40k Space Marines?)
Something to think about, at least, right?