I’m not sure what function you want to approximate. Is the 88-char-long seq your input or output? What should be the other one? If there is no input, are you trying to come up with a generative model for your sequences? Then look into things like GAN and VAE.
Using ordinal number as input is a bad idea, because the chars are not indications of some quantity. The usual way to deal with such data is one-hot-encoding, since the number of ASCII chars is small.
GANs, LSTMs, and MDNs are all different things. Among them, only LSTM is an architecture. You probably don’t want to use LSTM as the length of the sequence is fixed. GAN describes a generative network (regardless of architecture) whose loss function is defined by a simultaneously trained discriminator network. MDN describes a way to parametrize neural network output to represent a probability density function.