Do you need to use graphics engine? Not necessarily.

Should you use graphics engine? Yes I think you should.

Why? There are actually multiple reasons.
1. As you are being afraid using of standard LCL (Lazarus) or VCL (Delphi) components could become a problem in the future due to them not being able to render fast enough. Especially if you are planning on adding animations into your game. So by using graphical engine you can mostly avoid these problems. That is if you use it correctly. Even the best graphical engine can be brought to a crawl if not used properly.
2. By using of graphical engine you get access to some additional features that are not available with plain LCL or VCL. Most notable features that could come in handy for you are fast and smooth image scaling's, image rotations and fast multilayered rendering For instance you render background in one layer and then your units or buildings in another layer. Finally you combine these two layers together which can be very fast if they are both stored in video memory.
3. Since many graphical engines support multiple platforms using of such graphical engine means that you will have much easier job of porting your game to other platforms if you chose to do so. By using of LCL or VCL you are basically limited only on Windows.
4. Learning of working with graphical engine would help you in the long run, because I somehow doubt that this will be the only game you are going to try to make.

Now there are some drawbacks of moving to the usage of graphics engine.
1. Most notable drawback would be the fact that because many graphical engines does not work well in combination with LCL or VCL (could cause serious lags) you might be forced to recreate your whole UI with graphical engine instead of using LCL or VCL components.
But since you are making a turn based game where you don't need to continuously update game screen the chances for that are small.
2. Since rendering of images using graphical engine uses a different approach than using of LCL or VCL component is it might be a bit difficult to start as you also need to slightly adjust your thinking process regarding image rendering and scene building.

Any way I recommend you decide quick whether you will be using graphical engine or LCL/VCL. The main reason for this is that now when your game still isn't so complex the transition to usage of graphical engine would be much easier that later on when your game would become much more complex.

Now you are probably asking yourself: "If I chose to use a graphical engine which one should I use?"
I myself am interested in usage of Platform eXtended Library (http://asphyre.net/products/pxl) which is a very powerful graphical library but still fairly easy to use. I have used it predecessor (http://asphyre.net/products/legacy/9-asphyresphinx3) in the past and I must say that I have been very satisfied with it.
And thing that would probably make you most happy is that it is free to use even in a commercial product (it is distributed under Mozilla Public License (MPL) version 2 license).