First you are wrong, because you assume that all those are together, instead of looking as just a pairs of lion.

First find all possible combinations. Because all lions and tigers are distinct, then for the first place there are $(5+4)=9$ options, for the second $(5+4-1)=8$ options, and so on so the number of all posible combinations is:

$$9 \times 8 \times 7 \times \ldots \times 2 \times 1 = 9! = 362880$$

To get the exact solution you need to subtract the "bad" solution. So in order to find the bad solution, you could think of the 2 lions that are together as one. So there are 8 animals to rearrange. So there are $8! = 40320$.

Now we go back to the pair of lions. There are $5$ lions and there are $\binom{5}{2}$ ways to select a pair of lions so we have:

$$\binom 52 \times 8! = 403200$$

But that would give a negative amount, which is impossible. This happens because of double counting. Let the bold lions be pairs:

**$L, T, L, $ L, L$, T, T, L, T$**

**$L, T, $ L, L$, L, T, T, L, T$**

As you can in our cointing we count them as two, but they are actually one.

So in order to coun those "combinations" we would need to count special case, when we have 2 lions together, in between two tigers, then we have 3 lions together, in between two tigers. That's painful, but it get even more painful, because if those lions are at the end ot at the beggining they don't have to be in between two tigers. So I recommend you to stick to your first approach, because in combinatorics we can have multiple approaches, but we always should stick to the easiest one, that's your first approach.

You were lucky, because there is just one combination and that's:

$$L, T, L, T, L, T, L, T, L$$

If there were more you can add 2 "auxiliary" tigers, and then use the stars and bars calculation for $n-tuples$ with only positive integers. So we have $\binom{Tigers+2}{Lions}$ possible combinations. Then multiply by the number of permutations of the tigers and lions. So the final formula will look:

$$\binom{Tigers+2}{Lions} \times Tigers! \times Lions!$$