So, officially, the way to start with a higher-level character is to simulate them leveling up, basically. So you imagine this aasimar as a cleric 3rd, LA +1, and 6,000 XP when they perform the buy-off, costing 3,000 XP. That makes them a cleric 3rd, LA +0, and 3,000 XP.
Now, as you note, they gain more XP than their 4th-level allies, at least until they become 4th-level themselves—assuming they were fighting things of CR 3 or higher. Looking at the CR 3-CR 5 columns of the Table 2–6: Experience Point Awards (Single Monster) table on page 38 of the Dungeon Master’s Guide, a 3rd-level character earns ⅛ more than a 4th-level character for those encounters—good enough for me. That means that by the time they’ve earned 3,000 XP to reach 4th-level, their 4th-level allies have earned 3,000 XP × ⁸⁄₉ = 2,667 XP. Since everyone is 4th-level at this point, the remaining 1,333 XP needed for the other allies to reach the 10,000 XP they need for 5th level puts the aasimar at 7,333 XP to their allies’ 10,000 XP.
Coming to that answer required looking up two tables, in two separate books—neither of which was Unearthed Arcana—, and then performing my own back-calculation to come up with the ⅛ figure. In short, the Unearthed Arcana rules are simplistic and not full fleshed out—like much of that book, since it was just a collection of variant ideas that a DM could run with. There is reason to be wary here.
Which brings me to the more important point: you didn’t ask for the official way to handle it. You asked for a balanced way to handle it. And that, I’m afraid, no one can offer you: the entire “effective character level” system is broken, and cannot be fixed. The overwhelming majority of level adjustments are punitively high (most likely because Wizards of the Coast couldn’t afford to thoroughly test each one and so was conservative in how they assigned it), and even when they are, in some sense, “justified,” the resulting character is usually crippled anyway because no matter what overpowered feature they have, they also have far, far, too little hp, far too low saving throws, limited skill ranks, fewer feats, and on and on.
This is a well-known problem, and also a problem that Wizards of the Coast tacitly acknowledged as time went on—witness all of the myriad “LA-less” alternatives to these options that were published in later supplements. Player’s Guide to Faerûn even has a LA +0 “lesser aasimar” to deal with this (but it, unfortunately, is now overpowered). And either under- or over-powered, this becomes a headache for you as the DM, because it becomes harder to challenge the player appropriately.
Moreover, the annoyance I had calculating the XP for this aasimar? That doesn’t go away. You are going to be constantly having to figure out XP separately for the aasimar.
So you’ll be fighting this system both coming and going, when you design your encounters and then again when the players overcome them and are due XP. This is why I, and many others, outright refuse to allow split-XP parties in D&D 3.5e. It’s why Pathfinder, a 3.5e spin-off, removed XP costs from everything and eliminated the possibility altogether.
I strongly recommend that, instead of using the LA Buy-off rules, you just use some “lesser” aasimar for your games. Not the Player’s Guide to Faerûn one, though, since that’s blatantly overpowered (Wis +2, Cha +2, no penalties), but maybe use that as a starting point.
Or just give everyone in the party a free LA to play with. The aasimar player can have their aasimar, your other players can try some other higher-power race or grab an LA +1 template, whatever. This will still give you some headaches trying to design encounters, since they won’t really be as powerful as an extra level would be (at all) and you’ll have to figure something out, but then the CR rules aren’t (remotely) reliable enough where you could have avoided that kind of manual work anyway. At least their XP will stay consistent and they’re all relatively equal within the party. That makes things a whole lot easier.
And, as always, consider E6; many of these systemic problems are greatly exacerbated at 7th level and above, so avoiding those levels is quite useful.