If Feast said "burn this card, gain a card costing up to $5," and you Throned it, would you expect to gain two cards? I would like to say I feel like most people would, but actually "what happens when I Throne a Feast" used to be the most common rules question I was asked. Anyway I think I want you to still gain the second card there.
I think this post is the fundamental disagreement between the "Feast + ?" and "2xFeast" crowds regarding TR + BoM[Feast]. I also think the post is incorrect in the sense that a person undertaking a reasonable interpretation of the rules to date would not expect to gain two cards in this case.
The word "play" in throne room can be reasonably defined as "move this card from your hand to the play area. execute all text on it (the aforementioned card)," as that's what you do when you play an action normally. The second time Throne Room tries to "play" a card, it (the Throne Room) fails to move the card from the hand (it's already in the play area or somewhere else) but still executes text again. This is consistent with all previous interactions with released expansions (the Feast is executed again from the trash, so it gets you another card).
With that definition in hand, let's see what the interaction of Throne Room with BurnFeast is. To be clear, suppose "burning the card" means that there is no longer text on it (as it is now a pile of ashes). Now, when Throne Room tries to execute all text on the card the second time, it's been burnt, so we fail to do so, and we fail to gain our second card costing less than or equal to (5).
So let's analyze how this plays out with BoM. The text on BoM, as written, is equivalent to "When this enters play, replace all properties of it (name, cost, image*, artist*, type, text, expansion marker symbol*) with those of a card in the supply, but add 'When this card leaves play, it becomes a Band of Misfits**' to the text." Here, this interpretation is reasonable because that is exactly what this card does (becomes another one until it leaves play).
The current text of the cards, therefore, leads us to conclude that: first, TR + BoM[Smithy] leads us to drawing 6 cards (you choose Smithy when the card is first played, replacing its text with "+3 Cards; When this card leaves play, it becomes a Band of Misfits", which is executed twice successfully), and secondly, TR + BoM[Feast] choosing what target the second invocation of BoM has† as opposed to what TR + Feast would imply.
For the second half of your post -- "I think I want you to still gain the second card there" -- it's your game, and with it comes your design decisions. But I don't agree that this "rules interpretation" is something that is clear from previous ones and the text of the cards. If your interpretation is to survive, the text of Throne Room should be errata'd to be something that reflects your wish, as the only way I can see BurnFeast working out the way you claim it does is with some sort of memory attached to Throne Room (as written, "it" clearly refers to the 'card' that was chosen, not 'the text of the card when Throne Room resolves'). Contradictions with reasonable interpretations diminish gameplay experience -- ex falso quodlibet comes to mind for a reason why we'd want to minimize those.
*No revealed cards rely on these properties. But clearly they are copied anyway.
**Meaning, of course, those earlier mentioned properties are copied back onto the card from BoM
†The order of events that would happen is
1) Throne room is played with BoM as a target
2) BoM is played the first time, but its "Play this as if" text replaces the text on it with "Trash this card. Gain a card costing up to (5). When this card leaves play, it reverts to a BoM"
3) Text is executed. The BoM is trashed, you gain a card costing up to (5), and the text on it reverts to a BoM as it has left play.
4) Throne room plays the physical BoM card again. It fails to be removed from hand, but it gets its text executed. Its "play this as if" text triggers again, allowing the player to choose a second card.