How to completely redesign the site without losing positions in search engines?

Question

HTML

Hello!
TL;DR : We do not need website development services, thank you. :) We'll manage somehow.
There is a site of a small St. Petersburg organization. For those who immediately want to be horrified - link (PIK 99) .
The site stands on some kind of self-written CMS, which is provided by the hosting provider. Everything is organized in an incomprehensible way, it is quite difficult to find something. We connected an online store (an addition to the same CMS, from a hosting provider), as a result it turned out like two sites in one, one with general information about products, the other - a store. On the pages of the IM, the menu is duplicated: one is exclusively for the online store, the other is the old one, common for the site.
On the pages of IM brief information about the product and the opportunity to buy it. On regular pages - information about the same product, only without the ability to buy, only a link to the same product in the store. It would be nice to combine everything so that there is only a store with a full description, but this, like a number of other key problems, is difficult to do for a number of reasons. In other words, for one reason it is not possible to do the following:

• Remove all information about products not on the pages of the online store, having fully completed it;
  
• Redesign the site from scratch;
  
• Tear everything to hell, change hosting, install a normal well-thought-out open source CMS (the current one is closed) and make a site from scratch.
  

These and other possible implementations are not feasible because the site is well indexed and such changes will be fraught with a sharp drop in PS, but there is still some kind of conversion.
This raises one general question and several smaller ones.

1. How can you remake the site, make it an adequate look and normal structure, without losing positions?
   
2. Will changing the structure of the code while saving the content of the pages entail changes in the search results (for example, replacing the good old table structure with a layered one)?
   
3. In general, how do organizations deal with such situations when the site is cluttered, unstructured, but changes are fraught with unsuccessful reindexing?
   

Interested in your opinion, perhaps some other comments.
There is an opinion in the organization that such problems have a very negative impact on conversions, so we are wondering about changing the site.
In addition to general problems on the site, I would like to especially note the search, which is only on one page (the first menu item). CMS, it seems, does not allow you to place it in one place in the header.
There was an idea at first to at least make a more acceptable look, and, perhaps, if the answer to my second question is unsatisfactory, to resort exclusively to editing style files (at least somehow you can save the situation), but the site (like any other sample of 2006) , needless to say, drinks with inline styles like<p style="font-size: 14px"></p>, and there are not a few pages, the work will be very routine. It’s easier to do everything again (but you can’t do it again for the above reasons).
In general, it is very difficult to remake only the design, remaining on this management system: it is not logical in itself, and most of the site is not templates at all, but some kind of design made through the wysiwyg editor of static pages. Those. The amount of work is also overwhelming.
It is important to note that this site is the official site of the manufacturer.
Grateful for any hints!
Sincerely,
Alexander.

Answer 1

A lot of text...
1) Put redirects
2) Make a new site so that the URLs are saved
Please note that the redesign itself with the preservation of everything else can lead to a decrease in traffic from search engines.

Answer 2

I had such an order, I did not dare to do it myself. The man had an old site, in the first positions, he wanted to update the design without losing positions in search engines. Write to them yourself, they will advise you at [email protected]