Proving that the Minkowski metric tensor is invariant under Lorentz transformations

As you pointed out, the matrix form for the transformation equation of $eta$ is:

$$eta' = Lambda eta Lambda^T$$

It's possible to show that $eta' = eta$, given that $eta = Lambda^T eta Lambda$, recalling that $eta = eta^{-1}$:

$$begin{align} eta &= Lambda^T eta Lambda \ Lambda eta eta &= (Lambda eta Lambda^T) eta Lambda \ Lambda &=(Lambda eta Lambda^T) eta Lambda end{align}$$

This means that $Lambda eta Lambda^T = eta$ as well, so $eta = eta'$.

Alternatively, we can use the fact that $eta = Lambda^T eta Lambda$ must hold for all Lorentz transformations $Lambda$ and if $Lambda$ is a Lorentz transformation, then so is $Lambda^{-1}$. Taking the inverse of both sides:

$$begin{align} eta = Lambda^{-1} eta {(Lambda^{-1})}^T end{align}$$

However, what actually is $Lambda^{-1}$ is semantics since this equation holds for all $Lambda$ anyway, so we could just as well write $eta = Lambda eta Lambda^T$.

One final way to show that $eta' = eta$ is through the transformation of $eta_{mu nu}$ instead of $eta^{mu nu}$:

$$begin{align} eta'_{mu nu} &= Lambda^{alpha}{}_{mu} Lambda^{beta}{}_{nu} eta_{alpha beta} \ &= (Lambda^T)_{mu}{}^{alpha} eta_{alpha beta} Lambda^{beta}{}_{nu} end{align}$$

which has the direct matrix interpretation of $eta' = Lambda^T eta Lambda = eta$. This can then also be used to show why $Lambda^T eta Lambda = Lambda eta Lambda^T$.

Here's a no-nonsene approach to the question. The metric $eta$ is an object that acts on pairs of vectors $v, w$. In matrix notation, we can write that as $$eta(v, w) = v^{T} eta w$$ Now what happens if we apply the transformation $Lambda$ to these vectors? We get: $$eta(Lambda v, Lambda w) = (Lambda v)^{T} eta (Lambda w) = v^{T}(Lambda^{T} eta Lambda)w = v^T eta w = eta(v, w)$$ Hence the transformation $Lambda$ does preserve the metric. In fact it's not hard to convince yourself that $Lambda$ preserves the metric (for all possible pairs of vectors) if and only if the identity $eta = Lambda^t eta Lambda$ holds.

Now, with regards to your derivation, I suspect the problem is somewhere down the line your index notation is switching what should be lower indices with upper indices. So it's not surprising you end up transposed.

The Lorentz transformations are in particular coordinate transformations and the metric tensor is well, a tensor therefore it transforms as a tensor (left hand side of the following equation) and we demand it leaves the metric unchanged (right hand side of the equation): $$eta_{alphabeta} = Lambda^{mu}_{;;alpha} Lambda^{nu}_{;;beta}eta_{munu}tag{1}label{eq:lorentz}$$ or the contra-variant version of the metric if you will $$eta^{alphabeta} = Lambda^{alpha}_{;;mu} Lambda^{beta}_{;;nu}eta^{munu},$$ thus arriving to the defining property of the Lorentz transformations. The problems begin when you try to attach a matrix to the objects involved, so the transpose is to be taken with care. My suggestion is to stick to the tensor notation in realize a bit more geometrically what it means. We are looking for transformations that leave the space-time intervals unchanged: $$Delta s^2 = t^2 - vec{x}cdotvec{x} = x^mu, eta_{munu}, x^nu$$ Let us then explore what happens when we transform the coordinates according to a Lorentz transformation $$x'^mu = Lambda^mu_{;;nu},x^nu,tag{2}label{eq:coords}$$ by computing the space-time interval in these new coordinates: $$Delta s'^2 = x'^mu x'_mu = x'^mu eta_{munu} x'^nu = Lambda^mu_{;;beta}, x^beta, eta_{munu},Lambda^nu_{;;alpha}, x^alpha = x^beta (Lambda^mu_{;;beta},eta_{munu},Lambda^nu_{;;alpha})x^alpha$$ and now using the definition eqref{eq:lorentz} of the Lorentz transformations we get $$Delta s'^2 = x^beta eta_{betaalpha}x^alpha = Delta s^2$$ so it matches the interval in the original coordinates as we wanted.

For the sake of completeness if we have the convention that contra-variant (index up) means a column, then you can obtain/define the transpose of the coordinate transformation eqref{eq:coords}, by contracting with the metric and arrive to: $$ x'_mu = (eta_{mualpha}Lambda^alpha_{;;beta}eta^{betanu}),x_nuequiv x_nu (Lambda^T)_{;;mu}^{nu}$$ with this you can see that your property coincides with my equation eqref{eq:lorentz}.

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Not recommended for kids at home

To see how this is immediately your equation, remember that if you want to map these tensors into matrices, the first index is to be interpreted as pointing to the rows and the second index to the columns (up or down do not mean anything for this matter, just the order of appearance of the indices). So contracting the first indices of two objects is equivalent to contracting the second index of the first object transposed with the first index of the second object (same thing for sums where both indices are the second index). $$(A B)^{;,k}_{j} = A^{j}_{;i} B^{i k} = (A^T)^{i}_{;j} B^{i k} $$ notice that as matrices the above equation is ok, however the tensor structure is lost. Your equation is using this "matrix" map of the tensors, so when you write the metric tensor, the indices have a fixed meaning,

$$eta^{alphabeta} = Lambda^{alpha}_{;;mu} Lambda^{beta}_{;;nu}eta^{munu} = (Lambda^T)_{mu}^{;;alpha}Lambda^{beta}_{;;nu} eta^{munu}$$ so it means you have a matrix of rows indexed by $alpha$ and columns index by $beta$ the last expression just tells you how to compute it. That is we are adding a row against a row. For this reasons I don't recommend considering all tensors as matrices, a mixed (1,1) tensor, does have such natural interpretation but it is otherwise to be used with care. I really hope you see the issue is just a matter of trying to write down what is multiplied with what.