New PVP Stats = Missing Item Level?

Let's look at one example, both items are ilvl 476:

Dorian's Necklace of Burgeoning Dreams
849 sta + 566 int + 399 haste + 340 mastery = 2154

Malevolent Gladiator's Pendant of Alacrity
849 sta + 566 int + 405 haste + 331 mastery + (299 PvP Power + 299 PvP Resilience) = 2151 + (598) = 2749

As you can see, if you simply sum up all the stats without taking into account any sort of relative stat weights (which are irrelevant for this case as the existing PvE stats on both items are exactly the same) you can clearly see that they are pretty similar if you ignore the PvP stats (2154 vs 2151).
So we can conclude that PvP stats (Resilience and PvP Power) do not count towards the ilvl budget of any item.

So, yes even if PvP stats are out of the ilvl budget, the removal of resilience will still mean that PvP items are losing something extra that was exclusive to them, but PvP Power is still there and is still out of the budget, if we find that this is not enough we can always change some of the numbers on PvP items, like turning the PvP Power “knob” up until we are happy with them.

Note: Malevolent gear actually has even more stamina than that due to a recent hotfix, but I used the old values just to show that the numbers were correct in terms of correlation between stats and ilvl.

With respect but your comparison is misleading. You have to compare LFR items (ilvl502) with the recent honour gear. Random Battleground is the equivalent to Looking For Raid in terms of PvP. And thus there will be an imbalance. And this is definitely not from a PvP point of view.

How is it misleading? The main question was about whether or not resilience and PvP power counted towards the ilvl budget; this comparison was made to simply show that given two same ilvl and identical pieces, one PvE and one PvP, the PvE stats remain basically the same.

But is it intended that as a tank you have to obtain offspec PvP gear in order to be able to do Throne of Thunder LFR?

You don’t “have to”, but, you are absolutely free to do it.
That being said, I know where you are coming from and I mean, this is something that has been the case for ages, people like to take the quickest/shortest path available to them, and that often means going into arenas to grab some missing pieces solely for the ilvl so that you can meet some requirements, either to enter LFR or to be able to tell trade channel pugs that you meet the criteria of a certain average ilvl.
Now this might be slightly controversial, but I’d say that if you do that only for a few missing pieces, it won’t be much of a problem, LFR is, after all, a much easier version of raiding, and thus, even if you lack a bit of tanking stats, while not ideal, it won’t be disastrous, and you do still get the boost in HP.

So it’s really up to the player, if you can still perform well as a tank with less avoidance in LFR then there is no problem, but if you can’t, well, other players will probably notice it and you risk being kicked out of the raid group.

I feel like I have to mention this though, season 13 gear max ilvl is going to be 496 and 498 for weapons, and these take some time to get due to the existence of a conquest weekly cap (taking into account upcoming changes in 5.3).
If you go for Malevolent “honor” gear, then sure, you can get a full set in a week, but the current ilvl is 476, which isn’t high enough to queue for the latest LFR.
So you will still have to go through some of the lower LFR raids or get some PvE gear from Valor points to get into the top LFR at the moment.

Being able to choose is a good thing.
There have always been loopholes in complex systems, but I’d rather see players have the freedom to choose than restrict the game so much that we risk making PvE and PvP feel completely disconnected from each other, and while that is what a small group of players have always wanted, at the end of the day, we need to accept that this is still one game and one world, there needs to be a connection, else we could just have two completely separate games.

A solution that would be highly divisive but that would solve this sort of issue would be for example, make PvP gear non-equippable in PvE raids, and vice-versa. Would that be a good idea? Maybe for a niche, but personally I really wouldn’t like it and I don’t think it’s warranted, it’s not that big of a problem to make us resort to such drastic measures.

Having the option to use PvP gear in PvE and the opposite isn’t bad, it’s actually good, as long as we keep PvE gear the best for PvE content and PvP gear the best for PvP content, everything will work out fine.
I don’t think it would be nice to restrict someone that really enjoys PvP from entering the occasional LFR or normal raid just because his gear is only PvP, as long as his group is ok with it, what would be the problem? If a player dedicates himself mostly to PvP and spends some time and effort to get a full set of gear, and in the meantime masters his class on a PvP environment, then I believe that player has the minimum tools to be able to switch to PvE and perform at the very least as well as an average PvE player and vice-versa.
If that player then wants to get more serious into PvE, he will obviously have to invest in it and get PvE gear. I believe this is a good thing; it gives players more options without restricting access to any content.