swiftstrike has asi, which would help you if you fumble bombs and have to quickly swap to the right one. bombers also generally dont need much shielding since its far better to plant a bomb and NOT shield bump enemies away from the effect radius.

if you must get a 5* shield, i would recommend CoA, since devilite and greaver swarms can be pretty dangerous if you are caught off guard. it also has shock resist to stop that pesky status that prevents you from attacking.

does ASI have ANY affect on bombs whatsoever?!

That is just a really horrible recommendation. Maybe an experienced bomber can do that well, but the disadvantages of swiftstrike for a less experienced player are hardly worth the almost totally unnoticeable difference in bomb attack time.

Plus, you can just shield cancel if the 'putting the bomb back down' animation is bothering you.

Depends on what kind of attacks you find yourself most getting hit by and/or shield against. More specifically what kind of enemies.

For example, when I was getting into bombing my main problem was element damage. Gun Puppy boolits, random Scuttlebots that come out of nowhere; I thought I would have had more problems with fiends, but as a bomber I would have already blasted their tiny noggins off with some Barrage whereas with constructs and gremlins my first choice would have been Graviton. That meant getting closer than comfortable.

Plate shields are pretty good for defending against home run hitters. I heard Ancient Plate has more shield bumping force than the average shield.

Since bombers don't get any offensive bonuses from shields (other than Swiftstrike's ASI increase, which is negligible on a bomb), I would recommend you get a set of defensive shields.

Piercing: Aegis/Ironmight Plate - the first is cheap since its recipes are free and its 2* version is a mission reward, but with no status resist. The latter has stun resist and a lot of normal defence in exchange for some pierce defence, which is good in many situations; you'll need to buy its recipes, though, which can be found in HoH at standard prices.

Whatever anyone else tells you, do not bother with Barbarous Thorn Shield unless you plan on using swords as a sidearm. Otherwise, you will be paying considerably more for what is a (barely noticeably) weaker version of Aegis, since the sword bonus is useless if you're using pure bombs or bombs with guns. Boss reward recipes are only available from Basil, or from other players at often extortionate prices. And by 'extortionate' I mean that the 3* and 5* recipes can be found with a 10k cr price markup.

Elemental: GOwlite/Heater Shield - GOwlite protects against two of the more dangerous statuses, Shock and Fire, and makes an excellent shield in FSC as well as in general. Heater has more normal defence in exchange for elemental. However, it only protects against Freeze, which is rarely fatal, but gets a mention because evolves from the Defender line which makes it cheaper since only one recipe purchase is required and its 2* comes free. Both GOwlite and Heater Shield recipes are available in HoH.

Shadow: CoAlmire/Dread Skelly - Almire is a boss reward (30 Almirian Seals = possibly 10 Vanaduke runs) whose line runs only from 4* to 5*, and its one recipe can often be as expensive as BTS's 5* recipe. In return for your investment it protects against two of the more dangerous statuses, Shock and Fire, making it a great choice for FSC runs where shadow and fire are found in great quantities (although the resistance don't make any sense because its equivalent armour has fire weakness, and the Almirians are all meant to have burned to death) and in general. Dread Skelly has more shadow and less normal, and gets Freeze and Poison resist, which are somewhat less useful, but worthwhile in UGWW and HoI if you ever happen to do those. Its recipes are also freely available in HoH.

Ideally, you should have one shield for each specialised damage type, but if this is a secondary character and you're on a budget then going for the elemental option of GOwlite would probably be best. Constructs are extremely common, and it has defence against two dangerous statuses to boot.
(All of the above shields possess at least some level of normal defence, by the way.)

Other assorted shields:

• Omega - pure normal defence. Has a ton of health but lacklustre defence, and no status resistances. It's decent in t1 and not too bad in t2, but its usefulness is decreased somewhat in t3 when you're liable to encounter pure special damage from ranged attacks. Holds up reasonably against split normal-special attacks which occur in t2 and t3. Not especially great in comparison to the other pure normal shields, and made even worse by the fact you're going to have to put up with a tissue-paper shield until you get that Drake Scale up to heat level 5.
• Ancient Plate - pure normal defence and stun resist. Slightly less health than Omega, but higher normal defence in exchange. Stun resistance isn't particularly useful, either, and this is a boss token reward too (30 Almirian Seals).
• Volcanic Plate - pure normal defence, fire and stun resist. Great at tanking Vanaduke hammer slams, if you ever feel like doing so. Has about the same health and high normal defence as the Ancient Plate. Performs better than the other normal shields mentioned due to its fire resist, which is fairly useful in some Clockworks levels. Worth a look despite its lack of specialised defence.
• Dragon Scale - piercing/elemental defence, poison and fire resist. No normal defence means it breaks like tissue paper most of the time, since all attacks in t1 are pure normal and all of those in t2 have at least some modicum of normal damage to them. Excels in C42, but not elsewhere; elemental and piercing damage sources are rarely found together, and poison is generally one of the less dangerous statuses.
• Royal Jelly - normal/piercing defence and stun/sleep resistance, with higher pierce and lower normal defence than the Aegis. Since sleep sources were completely removed from the game (other than vials), 1/4 of the defensive power of this shield is completely redundant. It's best considered as being like Ironmight, but with much lower normal defence and not enough pierce defence to compensate for the loss. The only thing it has going for it is the fact you can get a Brute Jelly Shield for 15 Jelly Gems, meaning you can skip getting the first couple of recipes/buying a 3* shield.

tl;dr: Get defensive shields unless using sword sidearms regularly. Aegis and Heater Shield are cheaper but poorer options; GOwlite and Ironmight make better (if somewhat more expensive) alternatives. CoAlmire has more useful resistances than Dread Skelly, but is a pain to get. You might find Volcanic Plate to be useful. Only get BTS if you plan on using swords as sidearms.