Good question, I think there are several factors that are significant here that are worth discussing, starting with the two you mentioned in the question.
Tyler's feelings for Burnham
Of your two suggestions, I agree there's a good possibility Lorca is worried that Tyler's feelings for Burnham might compromise his decision-making abilities aboard the Klingon vessel. Of course, Lorca's not wrong in this assumption - Tyler and Burnham are barely aboard the Klingon vessel before Burnham has convinced Tyler to deviate from their mission to go rescue the Klingon's prisoner:
Burnham: I'm detecting a human life sign onboard.
Tyler: None of our intel on General Kol pointed to a potential Federation
Burnham: If we have a chance to rescue one of our people, we have to.
Tyler: Burnham, we can't deviate from the mission.
Burnham: The life sign isn't far from the bridge. Just on the opposite
side. Might even provide us a location to wait out the jumps. I have
to bring everyone home.
It doesn't take much at all to convince him, and that's without Burnham herself being placed in any real jeopardy. Of course, the big issue with this theory is that, if we accept Lorca has decided Burnham is more use on the Klingon ship than on the Discovery, why not have Tyler be the one to remain behind, thus removing the cause for his concern? Tyler can't be the only one on the Discovery capable of using a phaser, and it's not likely that Lorca was worried about sparing Tyler's blushes. Perhaps Burnham's plea convinced him that they would be a good team after all.
Burnham's value to the ship
I think Lorca would be less concerned about Burnham's perceived value to the ship. Not only have we seen on multiple occasions that Lorca is willing to sacrifice key players in order to win the war (first burning out the Tardigrade, then allowing the Admiral to be taken by the Klingons without attempting a rescue and finally allowing Stammets to push himself too far), but more than this, it's not her abilities as a "data-cruncher" that makes her valuable, it's her willingness to do what's right within the given context, even if that means disobeying Starfleet orders:
You chose to do the right thing, over and above what was sanctioned,
even at great cost to yourself. And that is the kind of thinking that
wins wars. The kind of thinking I need next to me. Universal law is
for lackeys. Context is for kings.
In addition to this, Burnham makes the case herself that she's there on borrowed time, and Lorca himself is ignoring a direct order to withdraw to the safety of Federation space. Worst case, none of them are coming back from the mission, but even if things go well, Lorca's gambling that Starfleet won't strip him of his commission and send Burnham back to prison immediately on his return. In that case, holding back a potentially advantageous player right now for the small possibility she might prove useful in the future doesn't really fit his character.
Lorca's fondness for Bunrham
It's possible Lorca has genuinely begun to look fondly on Burnham and wants to keep her from harm - in fact Burnham seems to play on this herself in her plea to him:
There is no logic to your thinking. Unless this is about me.
This might even be a factor in his his reasoning, but I don't think it would be his primary motivation. We saw he was willing to abandon Admiral Cornwell, so I'm not convinced he would let sentimentality affect his decision to forbid Burnham from going on the mission.
Burnham's history with the Klingons
A final possibility that's worth considering is that Lorca is concerned about Burnham's history with the Klingons and that particular ship and how they might use that history to manipulate her.
T'Kuvma killed Georgiou, Burnham's mentor and friend the last time she was on that ship, and again if this was Lorca's concern he was right on the money, as Kol instantly latches onto this fact when confronting Burnham.
Kol: And have you come here to take the place of the human that this
belongs to? [holds out Georgiou's insignia] I see that it holds value
to you. Well, it makes for a useful object to pick my teeth.
Kol also makes the very valid point that, should he kill her, the slayer of T'Kuvma, then one of the last Klingon holdouts to his supremacy, T'Kuvma loyalists, will come over to his side, increasing his power (and therefore his threat to the Federation).
Once I return with the slayer of their messiah, what remains of
T'Kuvma's torch-bearing fools will have no choice but to fall in line.
I'm not sure there's a way to definitively answer the question at this point, but if I had to take a guess, I'd say most of the above were likely factors to some extent, but that Tyler's relationship to Burnham and Burnham's history with the ship and the Klingons were likely uppermost in Lorca's mind.
I thought I should come back and update this in light of additional information we received toward the end of season one which drastically alters the way we might perceive this episode (so spoilers ahead, obviously).
Given we learn that Lorca is actually Lorca from the Mirror Universe, and that he needs Burnham as his ticket onto Mirror Universe Emperor Georgiou's ship to get to his crew and stage his coup, it's suddenly very apparent why he is reluctant to risk using her for this mission. It's not that she is valuable to the ship, she's actually pivotal to his own personal plan.
This also neatly explains why, given her value, he was still willing to risk her. He needed the Klingon mission to be a success because he needed the additional data Stamets generated during the micro-jumps in order to figure out how to return to the Mirror Universe. He must have realised that risking Burnham on the mission was the best way to ensure its success (if it failed, even if the Discovery managed to escape, he wouldn't have any way home and Burnham would no longer be as useful to him).