Can a monopoly INCREASE the market surplus compared with a competitive market?
Under standard assumptions (some of which you state in your question: no externalities, etc.), no. This follows from the First Welfare Theorem.

Perhaps there are departures from standard models that would support something resembling your conclusion, but my guess is that most economists would view any such departure as the absence of “perfect competition”.

It might be a good exercise for you to write this down in a model (even a simple one) to help you figure out what those departures might be.

Regarding the example you put in your edit:

First, your "monopoly" is not a monopoly in the sense that economists understand it. Monopolies have market power, and thus have no supply curve. The firm in your example has a supply curve, and has no market power.

Second, the example you give does not seem to all support your argument. The firms in the two different situations you present seem to have markedly different production technologies (reflected in the differing supply curves). Your argument requires that the firms have the same technology, only that perfect competition prevents the exploitation of "economies of scale" of their (common) technologies. In other words, this result is driven by differences in production technology, and not from the fact that one market is perfectly competitive and the other is monopolistic. (I believe this is what some comments to your question mean when they say that this is a weird comparison.)

Moreover, where exactly is economies of scale in the second situation? How am I supposed to be able to interpret what is going on as the firm being able to take advantage of economies of scale? Bear in mind that in the second situation, you would have the same outcome (with respect to surplus) if there were many firms, as long as the aggregate supply curve were the same. The fact that there is a single firm is not at all important to the result you've found.

Let's illustrate this previous argument specifically in your example. It seems the correct intervention for the state is not to allow the monopoly to continue but to regulate it. Instead, what the state should do is to make the monopolist's production technology available to other firms that want to enter the market to foster competition.

Suppose that the state does this, and another firm enters the market with the same technology. Let's also suppose now that the market is perfectly competitive. Each firm's supply curve must be

$$ q_s = -100 + 5P $$

which makes industry supply

$$ Q_s = -200 + 10P. $$

Given the demand curve $ Q_d = 1000 - P $, the equilibrium price and quantity is

$$ P^* = \frac{1200}{11}; \, Q^* = \frac{9800}{11}. $$

Total surplus is then $$ \frac{1}{2} (1000-20)\cdot \frac{9800}{11}, $$ a $7 \%$ increase in surplus. Surplus should also increase further when more firms enter.