When the concept of fluid antenna system (FAS) is applied to multiple-input multiple-output (MIMO) systems, this gives rise to MIMO-FAS, a.k.a.~fluid MIMO. Under rich scattering, the spatial correlation matrices are governed by the zeroth-order Bessel function $J_0(\cdot)$ through the continuously adjustable antenna positions, creating a highly non-convex landscape for optimization with fluctuating local optima -- the \emph{Bessel landscape}. In this paper, we tackle the joint transmitter (TX) and receiver (RX) antenna position optimization problem in fluid MIMO to maximize the ergodic capacity by shaping this landscape. Using Kronecker channel decomposition, we firstly develop a suite of analytical results that expose the problem's intrinsic structure: (i) a high signal-to-noise ratio (SNR) capacity approximation that decomposes the objective into separable log-determinant terms of the TX and RX correlation matrices, $\mathbf{R}_T$ and $\mathbf{R}_R$, respectively, (ii) a closed-form capacity loss bound linking $\det(\mathbf{R}_T)\det(\mathbf{R}_R)$ to the performance gap relative to the independent and identically distributed (i.i.d.) ideal MIMO channel, and (iii) the globally optimal inter-element spacing when the number of fluid elements at the TX is $N=2$ at the first zero of $J_0$. Guided by these insights, we propose two algorithms within an alternating optimization (AO) framework. The first algorithm is AO with particle swarm optimization (PSO) which deploys a particle swarm to explore the Bessel landscape globally without gradient information. Then in the second method, we use successive convex approximation (SCA) to obtain the gradient in closed form via $J_1(\cdot)$ to construct convex surrogates for orders-of-magnitude faster convergence.