Probably not with Seeming
A pegasus obviously doesn't have "The same basic arrangement of limbs" as a horse, so if it were going to work it would be leaning heavily on the equipment portion, creating an illusion of "horse wearing fake pegasus wings". While not definitively outside the scope of the spell, I probably would only allow it as a DM if you were at least aware of some specific fake pegasus wings to copy, or you were willing to give up on them moving. Requiring the same basic arrangement of limbs would be effectively a meaningless rule if players were allowed to have "a fake limb that just moves correctly at all times and is indistinguishable from a real limb" or "it has the right number of limbs but some of them are covered by an invisibility cloak". You can always ask your DM (I don't think it's overpowered or anything) but I wouldn't interpret the rules that way.
So what other options do you have?
Glyph of Warding
Major Image can easily handle the wings alone, even including appropriate sounds and smells, though probably not wind. If you wanted just a walking pegasus this would be trivial. Fly can give you a flying horse, and though it's not a Bard spell you already have that handled. The challenge is that both spells are concentration, and your party hates fun. Enter Glyph of Warding. Glyph of Warding has a bunch of text, but here is the relevant part with some emphasis added:
You can store a prepared spell of 3rd level or lower in the glyph by casting it as part of creating the glyph. The spell must target a single creature or an area. The spell being stored has no immediate effect when cast in this way. When the glyph is triggered, the stored spell is cast. If the spell has a target, it targets the creature that triggered the glyph. If the spell affects an area, the area is centered on that creature. If the spell summons hostile creatures or creates harmful objects or traps, they appear as close as possible to the intruder and attack it. If the spell requires concentration, it lasts until the end of its full duration.
Fly is a 3rd level spell which targets a single creature, so if you put it in a Glyph of Warding and have the horse trigger it, then the horse will get the full duration of Fly while you are free to concentrate on Major Image. This will fulfill all your conditions at least as well as your proposed use of Seeming + Fly using only 3rd level spells. Of course, there are a couple potentially major caveats: First, Glyph of Warding has a casting time of 1 hour, as opposed to the two actions that would be needed for just Fly and Seeming. Second, Glyph of Warding requires 200gp of material components, which could both strain your finances and attract attention in a different way. While this would give you 10 minutes of a very convincing fake pegasus it's definitely for special occasions only.
Polymorph
While regular Polymorph can't turn your steed into a Pegasus, both for CR reasons and because a Pegasus isn't a beast, it should be able to turn it into a different flying creature. Assuming your steed is a Warhorse from the Find Steed spell, D&D Beyond lists these valid Polymorph targets which are large enough to meet the basic requirement to serve as a mount for the typical adventurer. By far the most reasonable option there is the Giant Owl, which certainly isn't a Pegasus but would still be an impressive flying mount. Polymorph is a 4th level Bard spell and can last 1 hour with concentration, so this could even be maintained longer than Fly. You are subject to the DM ruling on whether an Owl has "appropriate anatomy" to serve as a mount though, so run it by them first.
Feather Fall
It requires some effort and setup, plus DM ruling on how momentum works if you want anything other than landing straight down, but Feather Fall could combine with Major Image for a "pegasus" gracefully coming to land. You would set up (presumably out of sight) by using Fly to reach 600 feet in the air, then drop concentration, immediately cast Feather Fall, then you are free to concentrate on Major Image. This is a fairly narrow way of achieving the effect (grand entrances only, pretty much) and a lot could go wrong since you won't have much control of the trajectory after you start the fall, but if you want people talking about "that impressive fellow that flew into town on a pegasus" as opposed to "that guy who can cast Fly" then it could do in a pinch.
Magic Items
In a comment you have clarified that magic items are an option. Interestingly, I haven't found many that seem like they would work, depending on your DM's interpretations. The most straightforward option is the Potion of Flying, since I don't think it's a stretch that a horse can drink a potion with a little help; this would provide a very similar effect to the Fly spell, which could then combine with Major Image. Potions are one-time use though, and going by official rarities a Potion of Flying seems like a hard thing to find. A more practical option would be the Winged Boots,
but I think it's a bit harder to sell both "Horse can wear boots" and "Horse can attune items". If they are allowed or a functionally-similar Horseshoe-type magic item can be arranged, then you could have your "pegasus" as often as you can cast Major Image. I was unable to find any magic items which provide the other half of the solution, either through transmutation or illusion, though I could be missing one.
