If we do not use path compression, we can maintain add[v] — the sum of all d added to the subtree of v. As the DSU are multiple trees, the weight of a vertex is the sum of add[p] for each parent p. Assume during merge we need to make parent[a] = b. We should make weights in a's subtree relevant by add[a] -= add[b].

If we want to use path compression, implementation becomes a bit more complicated. During find(int v) operation, we need to return not only the parent of v, but also the sum of add[u] for all u on path to the parent. If we know the sum, we can simply connect v directly to its parent and add the sum to add[v]. Obviously, all weights in v's subtree remain correct.

Your idea works, but with some minor modifications. For each component, store the vertices in that component as well (the time complexity is still $$$\mathcal{O}(n \log n)$$$ if you use small to large). Then, when merging component $$$X$$$ into component $$$Y$$$, we can first add the lazy value of component $$$X$$$ to all vertices in $$$X$$$, and subtract the lazy value of component $$$Y$$$ from all vertices in $$$X$$$, because when you are asked the value of a vertex $$$v$$$, you will output $$$\text{value}_v + \text{lazy}$$$ (where $$$\text{lazy}$$$ denotes the value of the lazy variable for the component of vertex $$$v$$$), but the lazy increment of the component $$$Y$$$ before you merged $$$X$$$ into it should not have any effect on the current vertex $$$X$$$, so in order to fix this mistake before we commit it, we will subtract $$$\text{lazy}_Y$$$ from the values of all the vertices of $$$X$$$ even before adding them to $$$Y$$$. Similarly, we should add $$$\text{lazy}_X$$$ to the values of all the vertices of $$$X$$$, because you should be adding that, but you don't (because the only thing you add right now is $$$\mathrm{lazy}_Y$$$), so in order to fix that mistake before we commit it, we will add $$$\text{lazy}_X$$$ to the values of all the vertices of $$$X$$$. The time complexity is $$$\mathcal{O}(n \log n)$$$ because of small to large.